Realization theory of discrete-time dynamical systems

front-matter......Page 1
1 Introduction......Page 11
2.1 Concatenation Monoid: Set of Experiments......Page 19
2.2 Input Response Maps......Page 20
2.3 Historical Notes and Concluding Remarks......Page 24
3.1 Realization Theory of General Dynamical Systems......Page 25
3.2 Finite General Dynamical Systems......Page 28
3.3 Control Systems and Multi-experiments......Page 33
3.4 Historical Notes and Concluding Remarks......Page 36
3.5.A U-Action......Page 38
3.5.B Pointed U-Actions......Page 40
3.5.C U-Actions with a Readout Map......Page 42
3.5.D General Dynamical Systems......Page 43
4.1! Realization!Theory!of!Linear!Representation!Systems......Page 48
4.2 Finite Dimensional Linear Representation Systems......Page 51
4.3 Partial Realization Theory of Linear Representation Systems......Page 58
4.4 Historical Notes and Concluding Remarks......Page 60
4 . 5 .A. 1 Linear State STRUCTURE: Linear -Actions......Page 62
4 . 5 .A. 2 Pointed Linear -Actions......Page 67
4 . 5 .A. 3 Linear U-Actions with a Readout Map......Page 69
4 . 5 .A. 4 Linear Representation Systems......Page 70
4 . 5 .A. 5 (Sophisticated) Realization Theorem......Page 71
4 . 5 . B . 1 Pointed Finite Dimensional Linear -Actions......Page 74
4 . 5 . B . 2 Finite Dimensional Linear -Actions with a Readout Map......Page 77
4 . 5 . B . 3 Finite Dimensional Linear Representation Systems......Page 79
4 . 5 . B . 4 Existence Criterion for Linear Representation Systems......Page 80
4 . 5 .C. 1 Pointed Linear U-Actions......Page 81
4 . 5 .C. 2 Linear U-Actions with a Readout Map......Page 82
4 . 5 .C. 3 Partial Realization Problem......Page 83
5.1 Realization Theorem of A0ne Dynamical Systems......Page 89
5.2 Finite Dimensional A0ne Dynamical Systems......Page 94
5.3 Historical Notes and Concluding Remarks......Page 101
5 . 4 .A. 1 Fundamentals for Affine Dyamical Stystems......Page 103
5 . 4 .A. 2 Affine State Structure: Linear U-Actions......Page 104
5 . 4 .A. 3 A;ne State Structure with a Readout Map......Page 108
5 . 4 .A. 4 A;ne Dynamical Systems......Page 111
5 . 4 .A. 5 Sophisticated Realization Theorem......Page 113
5 . 4 . B . 1 Pointed Finite Dimensional Linear U-Actions......Page 115
5 . 4 . B . 2 Finite Dimensional Linear -Actions with a Readout Map......Page 118
5 . 4 . B . 3 Finite Dimensional A;ne Dynamical Systems......Page 120
5 . 4 . B . 4 Existence Criterion for A;ne Dynamical Systems......Page 121
5 . 4 . B . 5 Realization Procedure for A;ne Dynamical Systems......Page 122
6.1 Input Response Maps with Time-Invariance......Page 123
6.2 Pseudo Linear Systems......Page 124
6.3 Finite Dimensional Pseudo Linear Systems......Page 127
6.4 Partial Realization Theory of Pseudo Linear Systems......Page 132
6.5 Real-Time Partial Realization Theory of Pseudo Linear System......Page 135
6.6 Historical Notes and Concluding Remarks......Page 137
6.7.A.1 Deriving Pseudo Linear Systems from A5ne Dynamical Systems......Page 138
6.7.A.2 Linear State Structure: Free Motions......Page 139
6.7.A.3 Free Motions with an A5ne Map......Page 142
6.7.A.4 Free Motions with a Readout Map......Page 145
6.7.A.5 Pseudo Linear Systems......Page 146
6.7.A.6 Sophisticated Pseudo Linear Systems......Page 148
6 . 7 . B . 1 Finite Dimensional Free Motions with an A5ne Map......Page 150
6 . 7 . B . 2 Finite Dimensional Free Motions with a Readout Map......Page 153
6 . 7 . B . 3 Finite Dimensional Pseudo Linear Systems......Page 155
6 . 7 . B . 5 Realization Procedure for Pseudo Linear Systems......Page 156
6 . 7 .C. 1 Free Motions with an A5ne Map......Page 157
6 . 7 .C. 2 Free Motions with a Readout Map......Page 158
6 . 7 .C. 3 Partial Realization Problem......Page 159
6.7.D Real-Time Partial Realization Problem......Page 164
7.1! Time-Invariant,!A5ne!Input!Response!Maps (Input!Response!Maps!with!Time-Invariance and Affinity)......Page 166
7.2 Almost Linear Systems......Page 167
7.3 Finite Dimensional Almost Linear Systems......Page 170
7.4 Partial Realization Theory of Almost Linear Systems......Page 175
7.5 Real-Time Partial Realization Theory of Almost Linear System......Page 177
7.6 Historical Notes and Concluding Remarks......Page 179
7 . 7 .A. 1 Derivation Almost Linear Systems from Pseudo Linear Systems......Page 181
7 . 7 .A. 2 Free Motions with an A4ne Map......Page 182
7 . 7 .A. 3 Free Motions with a Readout Map......Page 186
7 . 7 .A. 4 Almost Linear Systems......Page 188
7 . 7 .A. 5 Sophisticated Almost Linear Systems......Page 189
7 . 7 . B . 1 Finite Dimensional Free Motions with an A4ne Map......Page 191
7 . 7 . B . 2 Finite Dimensional Free Motions with a Readout Map......Page 194
7 . 7 . B . 3 Finite Dimensional Almost Linear Systems......Page 196
7 . 7 . B . 5 Realization Procedure for Almost Linear Systems......Page 197
7 . 7 .C. 1 Free Motions with an A4ne Map......Page 198
7 . 7 .C. 2 Free Motions with a Readout Map......Page 199
7 . 7 .C. 3 Partial Realization Problem......Page 200
7.7.D Real Time Partial Realization of Almost Linear Systems......Page 204
8 So-Called Linear Systems (Linear Systems with a Non-zero Initial State)......Page 205
8.1 Input Output Relations for So-Called Linear Systems......Page 206
8.2 So-Called Linear Systems and Almost Linear Systems......Page 207
8.3 Partial Realization Theory of So-Called Linear Systems......Page 213
8.4 Real-Time Partial Realization of So-Called Linear Systems......Page 217
8.5 Historical Notes and Concluding Remarks......Page 218
8.6.A So-Called Linear Systems and Almost Linear Systems......Page 220
8.6.B.1 Free Motions with an A1ne Map......Page 221
8.6.C Real-Time Partial Realization of So-Called Linear Systems......Page 227
back-matter......Page 228