Dynamical Modeling and Flatness Based Control of a Belt Drive System

Belt driven systems are part of many industrial applications, like computerized numerical control (CNC) machines in particular cutting machines and 3D-printers. In this paper the dynamical modeling and a flatness based controller design for belt driven systems are proposed. Due to the special kinematics, the stiffness of the belt is nonlinear, leading to nonlinear equations of motion. By neglecting some minor dynamical effects, the resulting system simplifies to a differentially flat one. This allows to calculate nominal feed-forward control torques by using the flat output of the system. To stabilize the error dynamics, an additional PD control law is introduced. The proposed method is compared with a controller, where elastic deflections for the feed forward part are neglected and elastic deformations are compensated by modifying the desired trajectories in a model-based manner. The tracking performance of both methods is evaluated in certain simulations and experiments. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rang des systèmes dynamiques gaussiens

Although some Gaussian dynamical systems, called “Gaussian-Kronecker”, share with rank one systems some unusual properties, we show in this work that a Gaussian dynamical system is never of finite rank. In fact, such a system can’t even be of local rank one.      

Tracing of pseudotrajectories of dynamical systems and stability of prolongations of orbits

We investigate properties of dynamical systems associated with the approximation of pseudotrajectories of a dynamical system by its trajectories. According to modern terminology, a property of this sort is called the “property of tracing pseudotrajectories” (also known in the English literature as the “shadowing property”). We prove that dynamical systems given by mappings of a compact set into itself and possessing this property are systems with stable prolongation of orbits. We construct examples of mappings of an interval into itself that prove that the inverse statement is not true, i.e., that dynamical systems with stable prolongation of orbits may not possess the property of tracing pseudotrajectories. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1016–1024, August, 1997.     

Boundary control and canonical realizations of a two-velocity dynamical system

The paper is devoted to the problems of controllability and realization for dynamical systems with various types of interacting waves that propagate with different velocities. One-velocity and a two-velocity dynamical systems are significantly different from the physical point of view. One can reconstruct a one-velocity system by its transfer function. For a two-velocity system a unique reconstruction is impossible. A procedure is proposed that allows us to construct by a transfer function of a two-velocity system a one-velocity system (a model) with the same transfer function. We give a “dynamical” interpretation for the triangular Krein factorization and for the corresponding construction of a triangular integral. For a transformation operator that connects a two-velocity system and its one-velocity model, a representation is given in terms of projectors on the accessible sets. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1994, pp. 18–44.     

An adaptive feedback control for chaos synchronization of nonlinear systems with different order

This paper deals with the chaotic synchronization of fourth-order system and second driven oscillators. Such a problem is related to the synchronization of strictly different chaotic systems. We show that the dynamical evolution of second-order driven oscillators can be synchronized with the projection on canonical planes of a fourth-order chaotic system. In this sense, it is said that the synchronization is achieved in reduced-order. Based on the Lyapunov approach, the adaptation law is determined to tune the controller gain vector in order to track a predetermined linearizing feedback control. An application to secure chaotic communication is also discussed. Numerical simulations are presented to demonstrate the efficiency of the proposed synchronization and secure communication schemes.     

Dynamical systems and simulation of turbulence

We propose an approach to the analysis of turbulent oscillations described by nonlinear boundary-value problems for partial differential equations. This approach is based on passing to a dynamical system of shifts along solutions and uses the notion of ideal turbulence (a mathematical phenomenon in which an attractor of an infinite-dimensional dynamical system is contained not in the phase space of the system but in a wider functional space and there are fractal or random functions among the attractor “points”). A scenario for ideal turbulence in systems with regular dynamics on an attractor is described; in this case, the space-time chaotization of a system (in particular, intermixing, self-stochasticity, and the cascade process of formation of structures) is due to the very complicated internal organization of attractor “points” (elements of a certain wider functional space). Such a scenario is realized in some idealized models of distributed systems of electrodynamics, acoustics, and radiophysics. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 217–230, February, 2007.     

Controllability in oscillation dynamical systems

We consider the problem of controllability in oscillation dynamical systems. A solution of the local control problem is obtained for one class of systems of differential equations. An example of application of the main results is given. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 183–191, February, 2008.     

On the Exel Crossed Product of Topological Covering Maps

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C ^*-algebras C(X)⋊ _α,ℒℕ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)⋊ _α,ℒℕ is a maximal abelian C ^*-subalgebra of C(X)⋊ _α,ℒℕ; any nontrivial two sided ideal of C(X)⋊ _α,ℒℕ has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ _α,ℒℕ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C ^*-algebras of homeomorphism dynamical systems.     

Destabilizing effect of random parametric perturbations of the white-noise type in some quasilinear continuous and discrete dynamical systems

We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester equations associated with stochastic Lyapunov functions of this form. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1719–1724, December, 2005.     

Number theoretical peculiarities in the dimension theory of dynamical systems

We show that dimensional theoretical properties of dynamical systems can considerably change because of number theoretical peculiarities of some parameter values. Supported by “DFG-Schwerpunktprogramm — Dynamik: Analysis, effiziente Simulation und Ergodentheorie”. We refer to the book of Falconer [6] for an introduction to dimension theory and recommend the book of Pesin [17] for the dimension theory of dynamical systems.     

Principal eigenvalues,topological pressure,and stochastic stability of equilibrium states

Suppose thatL is a second order elliptic differential operator on a manifoldM, B is a vector field, andV is a continuous function. The paper studies by probabilistic and dynamical systems means the behavior asɛ → 0 of the principal eigenvalueλ ^ε (V) for the operatorL ^ε = ɛL + (B, ∇) +V considered on a compact manifold or in a bounded domain with zero boundary conditions. Under certain hyperbolicity conditions on invariant sets of the dynamical system generated by the vector fieldB the limit asɛ → 0 of this principal eigenvalue turns out to be the topological pressure for some function. This gives a natural transition asɛ → 0 from Donsker-Varadhan’s variational formula for principal eigenvalues to the variational principle for the topological pressure and unifies previously separate results on random perturbations of dynamical systems. This work was supported by US-Israel Binational Science Foundation.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Order and Chaos in Dynamical Systems

We consider the characteristics of order and chaos in dynamical systems, with emphasis on the orbits in astronomical systems. Celestial mechanics deals with orbits in the solar system, which are mainly ordered. On the other hand the orbits of stars in galaxies were considered to be chaotic. However numerical experiments have shown that in general a system contains both ordered and chaotic orbits. Thus a new classification of dynamical systems has been established. We describe ordered and chaotic orbits in galaxies and in mappings. Some ordered orbits appear even in strongly perturbed systems. The transition from order to chaos is due to resonance overlapping. Then we describe some recent developments concerning order and chaos in the solar system and in galaxies. The outer spiral arms in strong barred galaxies are composed mainly of sticky chaotic orbits. Ordered and chaotic orbits appear also in Bohmian quantum mechanics. If the initial probability p is not equal to the square of the wave function |ψ|², then in the case of ordered orbits p never approaches |ψ|², while in the case of chaotic orbits p → |ψ|² after a time interval called “quantum Nekhoroshev time”.     

Rest points of generalized dynamical systems

In this paper we consider generalized dynamical systems whose integral vortex (that is, the set of all trajectories of the system starting at a given point) is an acyclic set in the corresponding space of curves. For such systems we apply the theory of fixed points for multi-valued maps in order to prove the existence of rest points. In this way we obtain new existence theorems for rest points of generalized dynamical systems. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 28–36, January, 1999.     

Technical stability of autonomous control systems with variable structure

We obtain conditions for the technical stability of autonomous dynamical systems with discontinuous control with respect to a given measure. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 12, pp. 1645–1658, December, 1999.     

Lowering topological entropy

The main result we prove in this paper is that for any finite dimensional dynamical system (with topological entropyh), and for any factor with strictly lower entropyh′, there exists an intermediate factor of entropyh″ for everyh″∈[h′, h]. Two examples, one of them minimal, show that this is not the case for infinite dimensional systems.     

Long-Time Behavior for Second Order Lattice Dynamical Systems

Many researchers examined the existence of global attractors for various types of first and second order lattice dynamical systems. Here we prove the existence of a global attractor for a new type of second order lattice dynamical systems in the Hilbert space l ²×l ². For specific choices of the linear operators this system can be regraded as a spatial discretization of a continuous damped nonlinear Boussinesq equation on ℝ ^m ,m≥1.      

On the linearization of isochronous centre of a modified Emden equation with linear external forcing

In this work, we carry out a detailed study on the linearization of isochronous centre of a modified Emden equation with linear external forcing. We construct inverse integrating factor and time independent first integral for this system through Darboux method. To linearize the isochronous centre we explore a transverse commuting dynamical system and its first integral. With the help of first integrals of the original dynamical system and its transverse commuting system we derive the linearizing transformation and reduce the nonlinear system into linear isochronous one. We also point out certain mathematical structures associated with this dynamical system.     

Universal minimal topological dynamical systems

Rokhlin (1963) showed that any aperiodic dynamical system with finite entropy admits a countable generating partition. Krieger (1970) showed that aperiodic ergodic systems with entropy < log a, admit a generating partition with no more than a sets. In Symbolic Dynamics terminology, these results can be phrased— ℕ^ℤ is a universal system in the category of aperiodic systems, and [a]^ℤ is a universal system for aperiodic ergodic systems with entropy < log a. Weiss ([We89], 1989) presented a Minimal system, on a Compact space (a subshift of ) which is universal for aperiodic systems. In this work we present a joint generalization of both results: given ɛ, there exists a minimal subshift of [a]^ℤ, universal for aperiodic ergodic systems with entropy < log a − ɛ.     

Dynamical Borel-Cantelli lemmas for gibbs measures

LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsA _n ⊃ X and μ-almost every pointx∈X the inclusionT ⁿ x∈A _n holds for infinitely manyn. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma. Partially supported by NSF grant DMS-9732728. Partially supported by NSF grant DMS-9704489.