Algebraic and Symbolic Computation Methods in Dynamical Systems

Preface......Page 7
Part II. Symbolic Methods for Nonlinear Dynamical Systems and for Applications to Observation and Estimation Problems......Page 8
Part III. Algebraic Geometry Methods for Systems and Control Theory......Page 9
Contents......Page 10
Part I Effective Algebraic Methods for Linear Functional Systems......Page 15
1 Effective Algebraic Analysis Approach  to Linear Systems over Ore Algebras......Page 16
1.1 Introduction......Page 17
1.2 Linear Systems over Ore Algebras......Page 20
1.3 Gröbner Basis Techniques......Page 28
1.3.1 Gröbner Bases for Ideals over Ore Algebras......Page 30
1.3.2 Gröbner Bases for Modules over Ore Algebras......Page 35
1.4.1 Linear Functional Systems and Finitely Presented Left Modules......Page 39
1.4.2 Basic Results of Homological Algebra......Page 44
1.4.3 Dictionary Between System Properties and Module Properties......Page 51
1.5.1 The HolonomicFunctions Package......Page 56
1.5.2 The OreAlgebraicAnalysis Package......Page 59
References......Page 63
2.1 Introduction......Page 66
2.2 Linear Functional Systems and Finitely Presented Left Modules......Page 68
2.3 Homomorphisms of Behaviors/Finitely Presented Left Modules......Page 74
2.4 Characterization of Isomorphic Modules......Page 81
2.5 The Unimodular Completion Problem......Page 91
References......Page 98
3 Computing Polynomial Solutions and Annihilators of Integro-Differential Operators with Polynomial Coefficients......Page 100
3.1 Introduction......Page 101
3.2 The Ring of Ordinary Integro-Differential Operators with Polynomial Coefficients......Page 102
3.3 Normal Forms......Page 105
3.4 Several Evaluations......Page 108
3.5 Syzygies and Annihilators......Page 110
3.6 Fredholm and Finite-Rank Operators......Page 114
3.7 Polynomial Solutions of Rational Indicial Maps  and Polynomial Index......Page 116
3.8 Polynomial Solutions and Annihilators......Page 122
References......Page 125
Part II Symbolic Methods for Nonlinear Dynamical Systems and for Applications to Observation and Estimation Problems......Page 128
4.1 Introduction......Page 129
4.2.1 Algebraic Systems......Page 131
4.2.2 Differential Systems......Page 137
4.3 Elimination......Page 145
4.4 Control-Theoretic Applications......Page 146
References......Page 156
5.1 Introduction......Page 159
5.2 The Differential Algebraic Approach......Page 160
5.3 How Does It Compare to the Classical Theory?......Page 161
5.4.1 Computing......Page 163
5.5 Regular Observability......Page 166
5.5.1 Sensor Selection......Page 167
5.6 Some of the Questions Without Partial Answers......Page 170
References......Page 171
6.1 Introduction......Page 173
6.2.1 Hereditary Theories......Page 175
6.2.2 Some Classical Integro-Differential Models......Page 176
6.3.1 Statement of the Estimation Problem......Page 178
6.3.2 The Algebraic Setting......Page 179
6.3.3 The Input–Output Equation of the Problem......Page 181
6.3.4 Algorithmic Transformation to Integro-Differential Form......Page 183
6.4 Towards Algebraic Theories......Page 184
6.4.1 Computational Issues......Page 186
6.4.2 On Generalizations of the Theorem of Zeros......Page 188
6.4.3 On Derivation-Free Elimination......Page 189
6.4.4 On Alternative Input–Output Equations......Page 190
References......Page 192
7.1 Introduction......Page 195
7.2 Problem Formulation......Page 197
7.2.1 Derivative Estimation Problem......Page 198
7.2.2 Parameter Estimation......Page 201
7.3 Annihilators via the Weyl Algebra......Page 202
7.4 Derivative Estimation......Page 204
7.5 Parameter Estimation......Page 205
7.7 Appendix......Page 208
References......Page 210
Part III Algebraic Geometry Methods for Systems and Control Theory......Page 213
8.1 Introduction......Page 214
8.2 Preliminaries......Page 216
8.3.1 GCD, Resultant, Subresultants......Page 219
8.3.2 Real Roots of Univariate Polynomials with Real Coefficients......Page 221
8.4 Gröbner Bases......Page 224
8.4.1 Applications of Gröbner Bases......Page 226
8.5.1 The Case of One Variable......Page 228
8.5.2 Univariate Representations of the Solutions......Page 230
8.5.3 Testing Structural Stability: The Zero-Dimensional Case......Page 235
8.6.1 Cylindrical Algebraic Decomposition......Page 238
8.6.2 Critical Point Methods......Page 243
References......Page 248
9.1 Introduction......Page 249
9.2 Problem Statement and Prerequisites......Page 251
9.3 Functional Birkhoff Matrices......Page 256
9.4.1 Recovering Polya-Segö Generic Bound......Page 257
9.4.2 On Beyond of Pólya-Szegő Bound......Page 260
9.5 Multiple Crossing Imaginary Roots with Non Zero Frequency......Page 261
9.6 Illustration on Inverted Pendulum: An Effective Approach versus Pólya-Szegő Bound......Page 263
References......Page 266
10 Controlled and Conditioned Invariance for Polynomial and Rational Feedback Systems......Page 269
10.2 Invariant Varieties of Autonomous Systems......Page 270
10.3 Controlled Invariant Varieties......Page 274
10.3.1 Nonuniqueness of Admissible Feedback Laws......Page 275
10.3.2 Rational Feedback......Page 278
10.4 Controlled and Conditioned Invariant Varieties......Page 283
10.4.1 Intersection of an Ideal and a Subalgebra......Page 284
10.4.2 Intersection of an Affine Ideal and a Subalgebra......Page 285
10.4.3 Rational Output Feedback......Page 290
References......Page 302
11.1 Introduction......Page 304
11.2 Preliminaries......Page 305
11.3 Control by Partial Interconnection......Page 307
11.4 Behavioral Controlled-Invariance......Page 309
11.4.1 The Case Where R  Is Full Row Rank......Page 310
11.4.2 The Case Where R  Is Not Full Row Rank......Page 313
11.5 Conclusions and Future Work......Page 314
References......Page 315
Index......Page 316