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E-grāmata: Algebraic Geometry For Robotics And Control Theory

(Univ Di Roma Tor Vergata, Italy), (Iasi-cnr, Italy), (Univ Di Roma Tor Vergata, Italy)
  • Formāts: 616 pages
  • Izdošanas datums: 02-Sep-2021
  • Izdevniecība: World Scientific Europe Ltd
  • Valoda: eng
  • ISBN-13: 9781800610477
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Algebraic Geometry For Robotics And Control TheoryPalielināt
  • Formāts: 616 pages
  • Izdošanas datums: 02-Sep-2021
  • Izdevniecība: World Scientific Europe Ltd
  • Valoda: eng
  • ISBN-13: 9781800610477
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"The development of inexpensive and fast computers, coupled with the discovery of efficient algorithms for dealing with polynomial equations, has enabled exciting new applications of algebraic geometry and commutative algebra. Algebraic Geometry for Robotics and Control Theory shows how tools borrowed from these two fields can be efficiently employed to solve relevant problem arising in robotics and control theory. After a brief introduction to various algebraic objects and techniques, the book first covers a wide variety of topics concerning control theory, robotics, and their applications. Specifically this book shows how these computational and theoretical methods can be coupled with classical control techniques to: solve the inverse kinematics of robotic arms; design observers for nonlinear systems; solve systems of polynomial equalities and inequalities; plan the motion of mobile robots; analyze Boolean networks; solve (possibly, multi-objective) optimization problems; characterize the robustness oflinear; time-invariant plants; and certify positivity of polynomials"--

The development of inexpensive and fast computers, coupled with the discovery of efficient algorithms for dealing with polynomial equations, has enabled exciting new applications of algebraic geometry and commutative algebra. Algebraic Geometry for Robotics and Control Theory shows how tools borrowed from these two fields can be efficiently employed to solve relevant problem arising in robotics and control theory.After a brief introduction to various algebraic objects and techniques, the book first covers a wide variety of topics concerning control theory, robotics, and their applications. Specifically this book shows how these computational and theoretical methods can be coupled with classical control techniques to: solve the inverse kinematics of robotic arms; design observers for nonlinear systems; solve systems of polynomial equalities and inequalities; plan the motion of mobile robots; analyze Boolean networks; solve (possibly, multi-objective) optimization problems; characterize the robustness of linear; time-invariant plants; and certify positivity of polynomials.
Preface v
About the Authors vii
List of Figures xv
List of Tables xix
List of Symbols xxi
1 Algebraic Geometry Notions 1(128)
1.1 Affine varieties and polynomial ideals
1(12)
1.2 Monomial orders and Grobner bases
13(16)
1.3 Homogeneous polynomials and ideals
29(4)
1.4 Elimination theory
33(12)
1.4.1 Resultants
44(1)
1.5 Operations on ideals
45(9)
1.6 Quotient rings and zero-dimensional ideals
54(25)
1.7 Root localization
79(24)
1.8 The Shape Lemma
103(7)
1.9 Solution of parametric systems for generic specializations of the parameters
110(6)
1.10 Modules and syzygies
116(13)
2 Implementations in Macaulay2 129(14)
2.1 The Macaulay2 language
129(3)
2.2 Polynomials in Macaulay2
132(1)
2.3 Monomial orders and Grobner bases
132(2)
2.4 Homogeneous polynomials and ideals
134(1)
2.5 Elimination theory
135(2)
2.6 Operations on ideals
137(1)
2.7 Quotient rings and zero-dimensional ideals
138(1)
2.8 The Shape Lemma
139(1)
2.9 Modules and syzygies
140(3)
3 The Inverse Kinematics of Robot Arms 143(22)
3.1 The direct kinematics
143(5)
3.2 The inverse kinematics
148(2)
3.3 Some arm-and-body structures
150(7)
3.3.1 The cylindrical robot
150(2)
3.3.2 The SCARA robot
152(2)
3.3.3 The spherical robot of Stanford
154(2)
3.3.4 The anthropomorphic robot
156(1)
3.4 The inverse orientation
157(8)
4 Observer Design 165(60)
4.1 Observability of real analytic and polynomial time-invariant systems
165(26)
4.2 Observability of time-varying systems
191(3)
4.3 Input-output embeddings of SISO continuous-time linear systems
194(4)
4.4 High-gain observers
198(4)
4.5 Parameter estimation
202(5)
4.6 Switching signal estimation
207(6)
4.7 Harmonic estimation for periodically forced chaotic systems
213(3)
4.8 Fault detection and isolation for a DC motor
216(9)
5 Immersions of Polynomial Systems into Linear Ones up to an Output Injection 225(18)
5.1 Output injection
225(2)
5.2 Immersion of polynomial systems into LIS form
227(5)
5.3 Immersion into LIS form up to a finite order
232(4)
5.4 Approximation of the immersion into MS form
236(7)
6 Solving Systems of Equations and Inequalities 243(58)
6.1 An algorithm to compute the solutions of systems of polynomial equations
243(5)
6.2 The method of the Lagrange multipliers
248(17)
6.3 Solution of systems of polynomial relations
265(7)
6.4 Application to the static output feedback stabilization problem
272(5)
6.5 Application to the stability analysis of planar polynomial systems
277(4)
6.6 Application to the dead-beat regulation of mechanical juggling systems
281(20)
6.6.1 Mechanical juggling systems
281(9)
6.6.2 Computation of the reference polynomial yd
290(11)
7 Motion Planning for Mobile Robots 301(40)
7.1 Well-defined affine varieties
301(3)
7.2 f-invariant affine varieties
304(18)
7.3 Locally attractive affine varieties
322(5)
7.4 Examples of f-invariant and attractive affine varieties
327(4)
7.5 Application to unicycle-like mobile robots
331(4)
7.6 Application to car-like mobile robots
335(6)
8 Computation of the Largest f-Invariant Set Contained in an Affine Variety 341(24)
8.1 f-invariant sets for continuous-time systems
341(11)
8.2 f-invariant sets for discrete-time systems
352(13)
9 Boolean Networks 365(32)
9.1 The Galois field F2
365(4)
9.2 Analysis of autonomous Boolean networks
369(13)
9.2.1 Reduced linear representation of Boolean networks
374(8)
9.3 Finite-horizon optimal control for Boolean networks
382(15)
9.3.1 Solution to integer programming problems
383(3)
9.3.2 Finite-horizon optimal control problem
386(1)
9.3.3 One-step optimization problem
387(10)
10 Multi-objective Optimization 397(42)
10.1 Multi-objective optimization in control system design
397(2)
10.1.1 Pole placement with compensators having a fixed structure
398(1)
10.1.2 Linear quadratic optimization
399(1)
10.2 Scalar optimization via algebraic geometry techniques
399(9)
10.2.1 Path-connected semi-algebraic sets
400(1)
10.2.2 Scalar minimization
400(7)
10.2.3 The envelope over an affine variety
407(1)
10.3 Multi-objective minimization
408(15)
10.3.1 The weighting method
410(1)
10.3.2 The method of rays
411(2)
10.3.3 The envelope method
413(2)
10.3.4 Test for the Pareto optimality
415(1)
10.3.5 Examples
416(7)
10.4 Solving control MOMPs
423(6)
10.4.1 Symbolic roots of a polynomial
424(3)
10.4.2 Reformulation of Problem 10.1.1
427(1)
10.4.3 Reformulation of Problem 10.1.2
428(1)
10.5 Application to physical plants
429(5)
10.5.1 Fast stabilization
429(1)
10.5.2 Pole placement
430(2)
10.5.3 Placement of the characteristic polynomial's coefficients
432(2)
10.5.4 Linear quadratic multi-objective optimization
434(1)
10.6 Further applications: Game design
434(5)
11 Distance to Internal Instability of Linear Time- Invariant Systems Under Structured Perturbations 439(60)
11.1 Introduction
439(2)
11.2 Related work in the unstructured case
441(2)
11.3 The border polynomial
443(17)
11.3.1 The continuous-time case
446(3)
11.3.2 The Sylvester matrix and the resultant
449(5)
11.3.3 The Bezout matrix and the resultant
454(3)
11.3.4 The discrete-time case
457(3)
11.4 Problem definition and first results
460(8)
11.5 The squared distance of a point to an affine variety
468(12)
11.6 The exponential stability in "almost all" cases
480(3)
11.7 Choosing the nominal point
483(9)
11.7.1 Analytic centers
484(5)
11.7.2 Chebyshev centers
489(3)
11.8 Control applications
492(7)
11.8.1 Continuous-time non-structured robustness analysis
492(1)
11.8.2 Discrete-time structured robustness analysis
493(2)
11.8.3 Parameter selection for a discrete-time system
495(2)
11.8.4 Optimal robust controller design
497(2)
12 Decomposition in Sum of Squares 499(74)
12.1 Introduction
499(8)
12.2 wSOS and the reduced echelon form
507(15)
12.2.1 Review of the quadratic case (d = 1): The "completing the square" procedure
508(6)
12.2.2 Definition of "generality" for wSOS
514(3)
12.2.3 A first solution to Problem 12.2.1 in the case n > m
517(5)
12.3 A certificate of positive (semi-)definiteness
522(2)
12.4 wS0S+ decomposition in the case n > m
524(21)
12.5 Examples
545(2)
12.6 Randomly generated experiments
547(4)
12.7 Applications in control and system theory
551(6)
12.8 wS0S+ decomposition through tools of linear algebra
557(16)
12.8.1 Saturation
558(6)
12.8.2 Polynomial representation of forms of total degree 2d
564(3)
12.8.3 Linear algebra implementation of the "completing the square" procedure
567(4)
12.8.4 wSOS+ representation
571(2)
Bibliography 573(12)
Index 585
Laura Menini received the Laurea degree in 1993 and her PhD in 1997 from the University of Rome Tor Vergata, where she is currently Full Professor. She co-authored the books Symmetries and Semi-invariants in the Analysis of Nonlinear Systems (Springer, 2011) and Mathematical Methods for System Theory (World Scientific, 1998). She is Chair of the IFAC Coordinating Committee 2 - Design Method and Senior Editor of the IEEE Control Systems Letters.

Corrado Possieri received his Bachelor's and Master's degrees in Medical Engineering and his PhD degree in Computer Science, Control and Geoinformation from the University of Rome Tor Vergata, Italy, in 2011, 2013, and 2016, respectively. In 2016, he visited the University of California, Santa Barbara (UCSB). In 2018, he joined the Dipartimento di Elettronica e Telecomunicazioni at the Politecnico di Torino, where he was an Assistant Professor. Subsequently, in 2019, he joined the Istituto di Analisi dei Sistemi ed Informatica "A Ruberti" of the Consiglio Nazionale delle Ricerche (IASI-CNR), where he is currently a Researcher. He is Associate Editor of the IEEE CSS Conference Editorial Board, the EUCA Conference Editorial Board, and the IFAC Conference Editorial Board.

Antonio Tornambč is currently Full Professor of Control Theory at the University of Rome Tor Vergata. His research interests are in control theory for linear and nonlinear systems. He is the co-author of Discrete-Event System Theory: An Introduction (World Scientific, 1995), Mathematical Methods for System Theory (World Scientific, 1998) and Symmetries and Semi-invariants in the Analysis of Nonlinear Systems (Springer, 2011).
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