There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

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Convex Analysisの二,三の進展について

The spatiotemporal expression of genes in an organism is determined by regulatory systems that involve a large number of genes connected through a complex network of interactions. As an intuitive understanding of the behavior of these systems is hard to obtain, computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This report reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, ordinary and partial differential equations, stochastic equations, Boolean networks and their generalizations, qualitative differential equations, and rule-based formalisms. In addition, the report discusses how these formalisms have been used in the modeling and simulation of regulatory systems.

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Modeling and simulation of genetic regulatory systems: a literature review.

A survey of models, analysis tools and compensation methods for the control of machines with friction

Dissipative Systems Analysis and Control

Dissipative Systems Analysis and Control (second edition) presents a fully revised and expanded treatment of dissipative systems theory, constituting a self-contained, advanced introduction for graduate students, researchers and practising engineers. It examines linear and nonlinear systems with examples of both in each chapter; some infinite-dimensional examples are also included. Throughout, emphasis is placed on the use of the dissipative properties of a system for the design of stable feedback control laws. The theory is substantiated by experimental results and by reference to its application in illustrative physical cases (Lagrangian and Hamiltonian systems and passivity-based and adaptive controllers are covered thoroughly). The second edition is substantially reorganized both to accommodate new material and to enhance its pedagogical properties. Some of the changes introduced are: * Complete proofs of the main theorems and lemmas. * The Kalman-Yakubovich-Popov Lemma for non-minimal realizations, singular systems, and discrete-time systems (linear and nonlinear). * Passivity of nonsmooth systems (differential inclusions, variational inequalities, Lagrangian systems with complementarity conditions). * Sections on optimal control and H-infinity theory. * An enlarged bibliography with more than 550 references, and an augmented index with more than 500 entries. * An improved appendix with introductions to viscosity solutions, Riccati equations and some useful matrix algebra.

Dissipative Systems Analysis and Control: Theory and Applications

1 Distributional model of impacts.- 1.1 External percussions.- 1.2 Measure differential equations.- 1.2.1 Some properties.- 1.2.2 Additional comments.- 1.3 Systems subject to unilateral constraints.- 1.3.1 General considerations.- 1.3.2 Flows with collisions.- 1.3.3 A system theoretical geometric approach.- 1.3.4 Descriptor variable systems.- 1.4 Changes of coordinates in MDEs.- 1.4.1 From measure to Caratheodory systems.- 1.4.2 Decoupling of the impulsive effects (commutativity conditions).- 1.4.3 From measure to Filippov's differential equations: the Zhuravlev-Ivanov method.- 2 Approximating problems.- 2.1 Simple examples.- 2.1.1 From elastic to hard impact.- 2.1.2 From damped to plastic impact.- 2.1.3 The general case.- 2.2 The method of penalizing functions.- 2.2.1 The elastic rebound case.- 2.2.2 A more general case.- 2.2.3 Uniqueness of solutions.- 3 Variational principles.- 3.1 Virtual displacements principle.- 3.2 Gauss' principle.- 3.2.1 Additional comments and studies.- 3.3 Lagrange's equations.- 3.4 External impulsive forces.- 3.4.1 Example: flexible joint manipulators.- 3.5 Hamilton's principle and unilateral constraints.- 3.5.1 Introduction.- 3.5.2 Modified set of curves.- 3.5.3 Modified Lagrangian function.- 3.5.4 Additional comments and studies.- 4 Two bodies colliding.- 4.1 Dynamical equations of two rigid bodies colliding.- 4.1.1 General considerations.- 4.1.2 Relationships between real-world and generalized normal di-rections.- 4.1.3 Dynamical equations at collision times.- 4.1.4 The percussion center.- 4.2 Percussion laws.- 4.2.1 Oblique percussions with friction between two bodies.- 4.2.2 Rigid body formulation: Brach's method.- 4.2.3 Additional comments and studies.- 4.2.4 Rigid body formulation: Fremond's approach.- 4.2.5 Dynamical equations during the collision process: Darboux-Keller's shock equations.- 4.2.6 Stronge's energetical coefficient.- 4.2.7 3 dimensional shocks- Ivanov's energetical coefficient.- 4.2.8 A third energetical coefficient.- 4.2.9 Additional comments and studies.- 4.2.10 Multiple micro-collisions phenomenon: towards a global coef-ficient.- 4.2.11 Conclusion.- 4.2.12 The Thomson and Tait formula.- 4.2.13 Graphical analysis of shock dynamics.- 4.2.14 Impacts in flexible structures.- 5 Multiconstraint nonsmooth dynamics.- 5.1 Introduction. Delassus' problem.- 5.2 Multiple impacts: the striking balls examples.- 5.3 Moreau's sweeping process.- 5.3.1 General formulation.- 5.3.2 Application to mechanical systems.- 5.3.3 Existential results.- 5.3.4 Shocks with friction.- 5.4 Complementarity formulations.- 5.4.1 General introduction to LCPs and Signorini's conditions.- 5.4.2 Linear Complementarity Problems.- 5.4.3 Relationships with quadratic problems.- 5.4.4 Linear complementarity systems.- 5.4.5 Additional comments and studies.- 5.5 The Painleve's example.- 5.5.1 Lecornu's frictional catastrophes.- 5.5.2 Conclusions.- 5.5.3 Additional comments and bibliography.- 5.6 Numerical analysis.- 5.6.1 General comments.- 5.6.2 Integration of penalized problems.- 5.6.3 Specific numerical algorithms.- 6 Generalized impacts.- 6.1 The frictionless case.- 6.1.1 About "complete" Newton's rules.- 6.2 The use of the kinetic metric.- 6.2.1 The kinetic energy loss at impact.- 6.3 Simple generalized impacts.- 6.3.1 2-dimensional lamina striking a plane.- 6.3.2 Shock of a particle against a pendulum.- 6.4 Multiple generalized impacts.- 6.4.1 The rocking block problem.- 6.5 General restitution rules for multiple impacts.- 6.5.1 Introduction.- 6.5.2 The rocking block example continued.- 6.5.3 Additional comments and studies.- 6.5.4 3-balls example continued.- 6.5.5 2-balls.- 6.5.6 Additional comments and studies.- 6.5.7 Summary of the main ideas.- 6.5.8 Collisions near singularities: additional comments.- 6.6 Constraints with Amontons-Coulomb friction.- 6.6.1 Lamina with friction.- 6.7 Additional comments and studies.- 7 Stability of nonsmooth dynamical systems.- 7.1 General stability concepts.- 7.1.1 Stability of measure differential equations.- 7.1.2 Stability of mechanical systems with unilateral constraints.- 7.1.3 Passivity of the collision mapping.- 7.1.4 Stability of the discrete dynamic equations.- 7.1.5 Impact oscillators.- 7.1.6 Conclusions.- 7.2 Grazing orC-bifurcations.- 7.2.1 The stroboscopic Poincare map discontinuities.- 7.2.2 The stroboscopic Poincare map around grazing-motions...- 7.2.3 Further comments and studies.- 7.3 Stability: from compliant to rigid models.- 7.3.1 System's dynamics.- 7.3.2 Lyapunov stability analysis.- 7.3.3 Analysis of quadratic stability conditions for large stiffness values.- 7.3.4 A stiffness independent convergence analysis.- 8 Feedback control.- 8.1 Controllability properties.- 8.2 Control of complete robotic tasks.- 8.2.1 Experimental control of the transition phase.- 8.2.2 The general control problem.- 8.3 Dynamic model.- 8.3.1 A general form of the dynamical system.- 8.3.2 The closed-loop formulation of the dynamics.- 8.3.3 Definition of the solutions.- 8.4 Stability analysis framework.- 8.5 A one degree-of-freedom example.- 8.5.1 Static state feedback (weakly stable task).- 8.5.2 Towards a strongly stable closed-loop scheme.- 8.5.3 Dynamic state feedback.- 8.6ndegree-of-freedom rigid manipulators.- 8.6.1 Integrable transformed velocities.- 8.6.2 Examples.- 8.6.3 Non-integrable transformed velocities: general case.- 8.6.4 Non-integrable transformed velocities: a strongly stable scheme.- 8.7 Complementary-slackness juggling systems.- 8.7.1 Some examples.- 8.7.2 Some controllability properties.- 8.7.3 Control design.- 8.7.4 Further comments.- 8.8 Systems with dynamic backlash.- 8.9 Bipedal locomotion.- A Schwartz' distributions.- A.1 The functional approach.- A.2 The sequential approach.- A.3 Notions of convergence.- B Measures and integrals.- C Functions of bounded variation in time.- C.1 Definition and generalities.- C.2 Spaces of functions of bounded variation.- C.3 Sobolev spaces.- D Elements of convex analysis.

Nonsmooth Mechanics: Models, Dynamics and Control

This book concerns the numerical simulation of dynamical systems whose trajectories may not be differentiable everywhere. They are named nonsmooth dynamical systems. They make an important class of systems, firstly because of the many applications in which nonsmooth models are useful, secondly because they give rise to new problems in various fields of science. Usually nonsmooth dynamical systems are represented as differential inclusions, complementarity systems, evolution variational inequalities, each of these classes being itself split into several subclasses. With detailed examples of multibody systems with contact, impact and friction and electrical circuits with piecewise linear and ideal components, the book is is mainly intended for researchers in Mechanics and Electrical Engineering, but it will be attractive to researchers from other scientific communities like Systems and Control, Robotics, Physics of Granular Media, Civil Engineering, Virtual Reality, Haptic Systems, Computer Graphics, etc.

https://hal.inria.fr/inria-00423530/PDF/book.pdf

Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics

Presents an adaptive control scheme for flexible joint robot manipulators. Asymptotic stability is insured regardless of the joint flexibility value, i.e., the results are not restricted to weak joint elasticity. Moreover, the joint flexibility is not assumed to be known. Joint position and velocity tracking errors are shown to converge to zero with all the signals in the system remaining bounded. >

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Bernard Brogliato

Papers

Dissipative Systems Analysis and Control

Dissipative Systems Analysis and Control: Theory and Applications

Nonsmooth Mechanics: Models, Dynamics and Control

Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics

Adaptive control of robot manipulators with flexible joints