Impulsive stabilization of mechanical systems in Takagi–Sugeno models

Takagi–Sugeno fuzzy impulsive systems are analyzed for Lyapunov stability. Lyapunov’s second method is used to establish sufficient stability conditions for such systems. It is shown that these conditions are expressed by a system of matrix inequalities. Impulsive fuzzy control of two coupled pendulums is considered as an example     

An enhanced input-delay approach to sampled-data stabilization of T–S fuzzy systems via mixed convex combination

The problem of sampled-data control is investigated for Takagi–Sugeno (T–S) fuzzy systems with aperiodic sampling intervals based on an enhanced input-delay approach. Delay-dependent stability and stabilizability conditions for the closed-loop continuous nonuniformly sampled-data fuzzy systems are derived by constructing a novel discontinuous Lyapunov–Krasovskii (L–K) functional, which makes good use of not only the upper bound on the variable sampling interval, but also its sawtooth structure information about varying input delay often ignored in previous results. A bounding technique combined by reciprocally convex technics and linear convex combination is presented for acquiring the time derivative of the functional, wherein Jensen’s inequality and Wirtinger’s inequality are integratively employed. And a feasible solution of the obtained criterion formulated as parameterized linear matrix inequalities is ultimately conceived. A numerical example is given to show the effectiveness of the proposed method.     

Pressure-hole errors—an alternative approach

A new analysis of the pressure-hole problem, which avoids some of the problems of the Higashitani-Pritchard (HP) analysis, is presented. The results are the same as those of the HP analysis for any fluid where the viscosity and the normal stress functions are multiples of one another. The relation of this result to experiments and computations is discussed.     

On the global stabilization of Takagi–Sugeno fuzzy cascaded systems

This paper deals with the problem of the global stabilization for a class of cascade nonlinear control systems. It is well known that, in general, the global asymptotic stability of the cascaded subsystems does not imply the global asymptotic stability of the composite closed-loop system. In this paper, we give additional sufficient conditions for the global stabilization of a cascade nonlinear system. In particular, we consider a class of Takagi–Sugeno (TS) fuzzy cascaded systems. Using the so-called parallel distributed compensation (PDC) controller, we prove that this class of systems can be globally asymptotically stable. An illustrative example is given to show the applicability of the main result.     

Udwadia–Kalaba constraint-based tracking control for artificial swarm mechanical systems: dynamic approach

Nonlinear Dynamics - A novel swarm tracking control for artificial swarm mechanical systems consisting of multiple mechanical agents is proposed. In this paper, the agents could not only perform...     

Global analysis of Ivlev＇s type predator-prey dynamic systems

Consider a class of Ivlev＇s type predator-prey dynamic systems with prey and predator both having linear density restricts. By using the qualitative methods of ODE, the global stability of positive equilibrium and existence and uniqueness of non-small amplitude stable limit cycle are obtained. Especially under certain conditions, it shows that existence and uniqueness of non-small amplitude stable limit cycle is equivalent to the local un-stability of positive equilibrium and the local stability of positive equilibrium implies its global stability. That is to say, the global dynamic of the system is entirely determined by the local stability of the positive equilibrium.     

Robust stability of uncertain piecewise-linear systems: LMI?approach

In this paper we propose sufficient conditions for the robust stability of time-invariant uncertain piecewise-linear systems using homogeneous polynomial Lyapunov functions. The proposed conditions are expressed in terms of linear matrix inequalities, which can be numerically determined. We solve the stabilization of piecewise uncertain linear control systems by using state piecewise-linear feedback. We propose an illustrative example to show the efficiency of the proposed approach.     

A structural mechanics approach for predicting the mechanical properties of carbon nanotubes

Based on molecular mechanics, a structural mechanics model of carbon nanotubes (CNTs) was developed with special consideration given to the bending stiffness of the graphite layer. The potentials associated with the atomic interactions within a CNT were evaluated by the strain energies of beam elements which serve as structural substitutions of covalent bonds in a CNT. In contrast to the original model developed by Li and Chou (Int. J. Solids Struct. 40(10):2487–2499, 2003), in the current model the out-of-plane deformation (inversion) of the bond was distinguished from the in-plane deformation by considering a rectangular cross-section for the beam element. Consequently, the model is able to study problems where the effect of local bending of the graphite layer in a carbon nanotube is significant. A closed-form solution of the sectional properties of the beam element was derived analytically. The model was verified through the analysis of rolling a graphite sheet into a carbon nanotube. Using the present model, the buckling behavior of nanotubes under bending is simulated. The predicted critical bending angle agrees well with molecular dynamics simulations.     

On application of Newton’s law to mechanical systems with motion constraints

This work is devoted to deriving and investigating conditions for the correct application of Newton’s law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton’s law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space and not on the dual space of a manifold, which is the natural geometrical space for this. Finally, the Euler–Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated and clarified further.     

A new Lyapunov approach for global synchronization of non-autonomous chaotic systems

This paper proposes a new approach for finding the Lyapunov function to study the sufficient global synchronization criterion of master-slave non-autonomous chaotic systems via linear state error feedback control. The approach is first demonstrated in a synchronization scheme for the second-order non-autonomous chaotic systems and then generalized to the schemes for the nth-order non-autonomous chaotic systems. Some algebraic synchronization criteria for the second-order chaotic systems are obtained. The sharpness of the new criteria is compared with that of the existing criteria of the same type by numerical examples.     

Intelligent quadratic optimal synchronization of?uncertain chaotic systems via LMI approach

This paper proposes an intelligent quadratic optimal control scheme via linear matrix inequality (LMI) approach for the synchronization of uncertain chaotic systems with both external disturbances and parametric perturbations. First, a four-layered neural fuzzy network (NFN) identifier is constructed to estimate system nonlinear dynamics. Based on the NFN identifier, an intelligent quadratic optimal controller is developed with robust hybrid control scheme, in which H _?? optimal control and variable structure control (VSC) are embedded to attenuate the effects of external disturbances and parametric perturbations. The adaptive tuning laws of network parameters are derived in the sense of the Lyapunov synthesis approach to ensure network convergence, and the sufficient criterion for existence of the controller is formulated in the linear matrix inequality (LMI) form to guarantee the quadratic optimal synchronization performance. Finally, a numerical simulation example is illustrated by the chaotic Chua??s circuit system to demonstrate the effectiveness of our scheme.     

Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.     

Floquet–Bloch decomposition for the computation of dispersion of two-dimensional periodic,damped mechanical systems

Floquet–Bloch theorem is widely applied for computing the dispersion properties of periodic structures, and for estimating their wave modes and group velocities. The theorem allows reducing computational costs through modeling of a representative cell, while providing a rigorous and well-posed spectral problem representing wave dispersion in undamped media. Most studies employ the Floquet–Bloch approach for the analysis of undamped systems, or for systems with simple damping models such as viscous or proportional damping. In this paper, an alternative formulation is proposed whereby wave heading and frequency are used to scan the k-space and estimate the dispersion properties. The considered approach lends itself to the analysis of periodic structures with complex damping configurations, resulting for example from active control schemes, the presence of damping materials, or the use of shunted piezoelectric patches. Examples on waveguides with various levels of damping illustrate the performance and the characteristics of the proposed approach, and provide insights into the properties of the obtained eigensolutions.     

Longitudinal dynamics of trains—a non-smooth approach

An efficient and straightforward algorithm for the integration of the constrained systems is developed in the present work. Some of the fundamental issues of contact dynamics are briefly reviewed. Considering Gauss?? least constraint principle, an intuitive and effective computation method of the constraint forces is arrived at. The contact problem is solved by means of the generalized matrix inverse (GI) embedded in an appropriate algorithm and the numerical integration is event-driven. The method is applied to the study of train longitudinal dynamics which is essential for comfort and running safety. Due to the complexity of the phenomena and the presence of dry friction, the investigations have been carried out by numerical methods using regularization. Based on the non-smooth dynamics approach, in the present paper, a comprehensive model is presented. Set-value friction of Coulomb??s law type is accounted for and motion equations are formulated as a differential inclusion. Application examples are presented. Simulations demonstrate the existence of specific phenomena like stick?Cslip and offset in the final equilibrium position. Computational efficiency is dramatically improved compared with previous models, while computation speed is increased by at least an order of magnitude. A wide class of problems may be solved using the present method, as is pointed out in the article.     

Path Integral approach via Laplace’s method of integration for nonstationary response of nonlinear systems

Meccanica - In this paper the nonstationary response of a class of nonlinear systems subject to broad-band stochastic excitations is examined. A version of the Path Integral (PI) approach is...     

Sliding mode control of a “Soft” 2-DOF Planar Pneumatic Manipulator

This paper presents a sliding mode controller for a “Soft” 2-DOF Planar Pneumatic Manipulator actuated by pleated pneumatic artificial muscle actuators. Since actuator dynamics is not negligible, an approximate model for pressure dynamics was taken into account, which made it necessary to perform full input-output feedback linearization in order to design a sliding mode controller. The design of the controller is presented in detail, and experimental results obtained by implementing the controller are discussed Published in Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 135–144, October 2008.     

Pointwise-in-space stabilization and synchronization of a class of reaction–diffusion systems with mixed time delays via aperiodically impulsive control

Nonlinear Dynamics - This paper investigates the problems of pointwise-in-space stabilization and synchronization of semilinear reaction–diffusion systems with Dirichlet boundary conditions...