A note on Poisson stability and controllability

The connection between Poisson stability and controllability is revisited. For the controllability criterion based on weakly positive Poisson stability (WPPS) and Lie algebra rank condition (LARC), it is shown that two natural assumptions about the set of input values are necessary for strictness. By using the relationship between conservative property and WPPS, sufficient conditions for controllability of affine nonlinear systems are further discussed. At last, for systems defined on compact Riemannian manifold, the WPPS is proved to be equivalent to Poisson stability.     

Optimal control of quadratic functionals for affine nonlinear systems

In this paper we analyze the optimal control problem for a class of affine nonlinear systems under the assumption that the associated Lie algebra is nilpotent. The Lie brackets generated by the vector fields which define the nonlinear system represent a remarkable mathematical instrument for the control of affine systems. We determine the optimal control which corresponds to the nilpotent operator of the first order. In particular, we obtain the control that minimizes the energy of the given nonlinear system. Applications of this control to bilinear systems with first order nilpotent operator are considered.     

Controllability of delay degenerate control systems with independent subsystems

IntroductionAsthestudyforthetheoryofcontrolsystemsdeeplygoesonandtheneedforthestudyandtheapplicationofmanypracticalsystemssuchaspowersystems,ecosystems,economicmanagementsystemsandsoon ,peoplerequestthattheprecisionforthedescribing ,analysisanddesignabout…     

Homogeneous flow field effect on the control of Maxwell materials

The controllability of viscoelastic fields is a fundamental concept that defines some essential capabilities and limitations of the resulting materials. In this paper, we study the controllability of different homogeneous flow fields of viscoelastic fluids governed by the upper convected Maxwell model. The approach is largely based on the nonlinear geometric control theory. Through the analysis of the control Lie algebra, we find the submanifolds in the state space on which the homogeneous flow fields are weakly controllable. Our approach can be generalized to more complicated systems.     

Controllability of an underactuated spacecraft with one thruster under disturbance

For an underactuated spacecraft using only one thruster,the attitude controllability with respect to the orbit frame is studied in the presence of periodical oscillation disturbance,which provides a preconditional guide on designing control law for underactuated attitude control system.Firstly,attitude dynamic model was established for an underactuated spacecraft,and attitude motion was described using the special orthogonal group(SO(3)).Secondly,Liouville theorem was used to confirm that the flow generated by the drift vector of the underactuated attitude control system is volume-preserving.Furthermore,according to Poincare’’s recurrence theorem,we draw conclusions that this drift field is weakly positively poisson stable(WPPS).Thirdly,the sufficient and necessary condition of controllability was obtained on the basis of lie algebra rank condition(LARC).Finally,the controllable conditions were analyzed and simulated in different cases of inertia matrix with the installed position of thruster.     

The invariant manifold method and the controllability of nonlinear control system

The problem of controllability of nonlinear control system is a significant field which has an extensive prospect of application. A.M.Kovalev of Ukraine Academy of Science applied the oriented manifold method developed in dynamics of rigid body to nonlinear control system for the first time and obtained a series of efficient results. Based on Kovalev’s oriented manifold method, firstly, by invariant manifold method the problem of controllability of nonlinear control system was studied and the necessary condition of the controllability of a kind of affine nonlinear system was given out. Then the realization of the necessary condition was discussed. At last, the motion of a rigid body with two rotors was investigated and the necessary condition which is satisfied by this system was proved.     

Global controllability of certain systems of rigid bodies under friction

We consider Lagrangian systems with bounded potential energy, with control constraints known in advance, and with friction. We present several sufficient conditions for the global controllability in the case of a cyclic coordinate assuming that the dry friction forces acting in the system satisfy the Amonton-Coulomb law. In particular, we prove the global controllability of a system of three equal masses successively connected by linear springs on a common rough line when the only controlling action is a bounded force applied to the first load. A majority of well-known results concerning controllability (in the traditional sense [1]) of dynamical systems were obtained only for the cases in which the continuity of the right-hand side of the differential equation is either assumed in advance or is ensured by part of the controlling resources. In the present paper, we give several sufficient conditions for the global controllability of systems of rigid bodies under the action of dry friction forces; the number of controls can be less than the number of degrees of freedom. The main idea of our approach consists in searching and using auxiliary “attainable” curves in the phase space such that these curves lie outside the domains of “nonsmoothness” of the system.     

Navier–Stokes Equations: Controllability by Means of Low Modes Forcing

We study controllability issues for 2D and 3D Navier–Stokes (NS) systems with periodic boundary conditions. The systems are controlled by a degenerate (applied to few low modes) forcing. Methods of differential geometric/Lie algebraic control theory are used to establish global controllability of finite-dimensional Galerkin approximations of 2D and 3D NS and Euler systems, global controllability in finite-dimensional projection of 2D NS system and L2-approximate controllability for 2D NS system. Beyond these main goals we obtain results on boundedness and continuous dependence of trajectories of 2D NS system on degenerate forcing, when the space of forcings is endowed with so called relaxation metric.     

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅱ)—SINGLE TIME_DELAY CASE

IntroductionConsideraswitchedlinearsystemwithsingletime_delayinthecontrolfunctiongivenby x(t) =Ar(t) x(t) +Br(t) u(t) +Dr(t) u(t -τ) ,(1)wherex(t) ∈Rnisthestate,u(t) ∈R mtheinput,thepiecewiseconstantscalarfunctionr(t) :R+ → 1,2 ,… ,Nistheswitchingpath ,(Ai,Bi,Di) ｜i =1,… ,Nisafinitefamilyofsystemrealizations.Moreover,r(t) =iimpliesthat(Ai,Bi,Di)ischosenasthesystemrealizationattimet.τ >0isthefixedtimedelayincontrol.Inthispaper ,wewillstudythecontrollabilityofsystem (1) .Theone_per…     

亏损系统振动模态的可控可观性

利用奇异值分解方法来讨论系统广义模态的可控可观性的量度问题,得到了亏损系统广义模态可控可观性的量度指标,同时用实例说明了本文方法是有效的。     

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅲ)—MULTIPLE TIME_DELAY CASE

IntroductionConsidertheswitchedlinearsystemwithmultipletime_delaysincontrolfunctiongivenby x(t) =Ar(t) x(t) +Br(t) u(t) + Kk=1Dr(t) ,ku(t -τk) ,(1)wherex(t) ,u(t)andr(t)isdefinedasbefore ,(Ai,Bi,Di,1,… ,Di,K)｜i=1,… ,Nisafinitefamilyofsystemrealizations.Moreover,r(t) =iimplies (Ai,Bi,Di,1,… ,Di,K)ischosenasthesystemrealizationattimet.K <∞isthenumberoftime_delaysinthesystem .0<τ1<… <τKareKfixedtime_delays .Remark 1　Similarly ,wecandescribeswitchingpathbyintroducingswitchingsequ…     

Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations

Conditions are derived for the linearizability via invertible maps of a system of n second-order quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, ℝ). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two- and three-dimensional systems.     

Modeling and analysis of nonholonomic dynamic systems with a class of rheonomous affine constraints

This paper is devoted to modeling and theoretical analysis of dynamic control systems subject to a class of rheonomous affine constraints, which are called $A$ -rheonomous affine constraints. We first define $A$ -rheonomous affine constraints and explain their geometric representation. Next, a necessary and sufficient condition for complete nonholonomicity of $A$ -rheonomous affine constraints is shown. Then, we derive nonholonomic dynamic systems with $A$ -rheonomous affine constraints (NDSARAC), which are included in the class of nonlinear control systems. We also analyze linear approximated systems and accessibility for the NDSARAC. Finally, the results are applied to some physical examples in order to check the application potentiality.     

新型Poisson括号意义下的无穷维Lie代数

本文首先针对ＫｄＶ方程的Ｈａｍｉｌｔｏｎ形式，建立一种比较容易验证的新型Ｐｏｉｓｓｏｎ括号和无穷维Ｌｉｅ代数．其次，研究ＫｄＶ方程的Ｈａｍｉｌｔｏｎ形式的第一积分与新型Ｐｏｉｓｏｎ括号的关系，得到判定第一积分的充分必要条件．最后，构造ＫｄＶ方程的第一积分．     

Constrained controllability of semilinear systems with delays

In the paper, finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point time-variable delays in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local relative controllability are formulated and proved. It is generally assumed that the values of admissible controls are in a convex and closed cone with the vertex at zero. The special case of constant multiple point delays is also discussed. Moreover, some remarks and comments on the existing results for controllability of nonlinear and semilinear dynamical systems are also presented.     

Reduction of some classical non-holonomic systems with symmetry

Two types of nonholonomic systems with symmetry are treated: (i) the configuration space is a total space of a G-principal bundle and the constraints are given by a connection; (ii) the configuration space is G itself and the constraints are given by left-invariant forms. The proofs are based on the method of quasicoordinates. In passing, a derivation of the Maurer-Cartan equations for Lie groups is obtained. Simple examples are given to illustrate the algorithmical character of the main results.     

On the van der Waals interaction of carbon nanotubes as electromechanical nanothermometers

Electromechanical carbon nanothermometers are devices that work based on the interactions and relative motions of double-walled carbon nanotubes (DWCNTs). In this paper, the mechanics of carbon nanotubes (CNTs) constituting two well-known configurations for nanothermometer, namely shuttle configuration and telescope configuration are fully investigated. Lennard-Jones (LJ) potential function along with the continuum approximation is employed to investigate van der Waals (vdW) interactions between the interacting entities. Accordingly, semi-analytical expressions in terms of single integrals are obtained for vdW interactions. Acceptance condition and suction energy are studied for the shuttle configuration. In addition, a universal potential energy is presented for the shuttle configuration consisting of two finite CNTs. Also, for the telescope configuration, extensive studies are performed on the distributions of potential energy and interaction force for various radii and lengths of CNTs. It is found that these geometrical parameters have a considerable effect on the potential energy.     

Explicit solutions for a SWCNT collapse

Self-collapse of carbon nanotubes may prominently affect their electrical properties and holds promise for important applications in the emerging nano-mechanical or electronic systems. Based upon the potential energy functional, we derived the governing equation and transversality boundary condition for a collapsed single-walled carbon nanotube (SWCNT). Considering the inextensible condition of the elastica, some closed-form solutions for the collapsed configuration were obtained in terms of elliptical integrals. These analytical solutions include the critical length of the flat contact segment, critical radii for collapsed and circular shapes of the cross-section, deflections and potential energies of the SWCNTs. Finally, the energy states and the collapsed morphologies of the SWCNTs were presented. These explicit solutions are beneficial for the design of nano-structured materials, and cast a light on enhancing their mechanical, chemical, optical and electronic properties.     

On the general form of the law of dynamics

The dynamics of Lagrangian systems is formulated with a differential geometric approach and according to a new paradigm of the calculus of variations. Discontinuities in the trajectory, non-potential force systems and linear constraints are taken into account with a coordinate-free treatment. The law of dynamics, characterizing the trajectory in a general non-linear configuration manifold, is expressed in terms of a variational principle and of differential and jump conditions. By endowing the configuration manifold with a connection, the general law is shown to be tensorial in the velocity of virtual flows and to depend on the torsion of the connection. This result provides a general expression of the EULER-LAGRANGE operator. POINCARÉ and LAGRANGE forms of the law are recovered as special cases corresponding, respectively, to the connection induced by natural and mobile reference frames. For free motions, the geodesic property of the trajectory is directly inferred by adopting the LEVI-CIVITA connection induced by the kinetic energy.     

Dissipativity-based stable controller redesign for nonlinear MIMO switched systems in the presence of perturbations

This paper offers a control redesign method for a class of nonlinear MIMO perturbed dissipative switched systems. The controller for the nominal switched system without perturbation is considered to have been designed based on dissipativity. The nominal model is considered affine with perturbations in both nonlinear terms. The controller is then redesigned such that the dissipativity of the switched system with perturbations is maintained by including a robustifying term in the control law. The design is developed for the two cases of 2-norm and infinity-norm bounded perturbations. The proposed method is applied to a nonlinear MIMO switched system to verify the convergence of the state vector to the origin in the presence of perturbations.