Balanced Truncation Model Reduction of a Nonlinear Cable-Mass PDE System with Interior Damping

We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters

Synchronization of coupled stochastic limit cycle oscillators

For a class of coupled limit cycle oscillators, we give a condition on a linear coupling operator that is necessary and sufficient for exponential stability of the synchronous solution. We show that with certain modifications our method of analysis applies to networks with partial, time-dependent, and nonlinear coupling schemes, as well as to ensembles of local systems with nonperiodic attractors. We also study robustness of synchrony to noise. To this end, we analytically estimate the degree of coherence of the network oscillations in the presence of noise. Our estimate of coherence highlights the main ingredients of stochastic stability of the synchronous regime. In particular, it quantifies the contribution of the network topology. The estimate of coherence for the randomly perturbed network can be used as means for analytic inference of degree of stability of the synchronous solution of the unperturbed deterministic network. Furthermore, we show that in large networks, the effects of noise on the dynamics of each oscillator can be effectively controlled by varying the strength of coupling, which provides a powerful mechanism of denoising. This suggests that the organization of oscillators in a coupled network may play an important role in maintaining robust oscillations in random environment. The analysis is complemented with the results of numerical simulations of a neuronal network. PACS: 05.45.Xt, 05.40.Ca Keywords: synchronization, coupled oscillators, denoising, robustness to noise, compartmental modelComment: major revisions; two new section

Lyapunov functions for linear damped wave equations in one-dimensional space with dynamic boundary conditions

We establish the exponential decay of the solutions of the damped wave equations in one-dimensional space where the damping coefficient is a nowhere-vanishing function of space. The considered PDE is associated with several dynamic boundary conditions, also referred to as Wentzell/Ventzel boundary conditions in the literature. The analysis is based on the determination of appropriate Lyapunov functions and some further analysis. This result is associated with a regulation problem inspired by a real experiment with a proportional-integral control. Some numerical simulations and additional results on closed wave equations are also provided

Robust error estimates in weak norms for advection dominated transport problems with rough data

We consider mixing problems in the form of transient convection--diffusion equations with a velocity vector field with multiscale character and rough data. We assume that the velocity field has two scales, a coarse scale with slow spatial variation, which is responsible for advective transport and a fine scale with small amplitude that contributes to the mixing. For this problem we consider the estimation of filtered error quantities for solutions computed using a finite element method with symmetric stabilization. A posteriori error estimates and a priori error estimates are derived using the multiscale decomposition of the advective velocity to improve stability. All estimates are independent both of the P\'eclet number and of the regularity of the exact solution

Numerical computation of complex multi-body Navier-Stokes flows with applications for the integrated Space Shuttle launch vehicle

An enhanced grid system for the Space Shuttle Orbiter was built by integrating CAD definitions from several sources and then generating the surface and volume grids. The new grid system contains geometric components not modeled previously plus significant enhancements on geometry that has been modeled in the old grid system. The new orbiter grids were then integrated with new grids for the rest of the launch vehicle. Enhancements were made to the hyperbolic grid generator HYPGEN and new tools for grid projection, manipulation, and modification, Cartesian box grid and far field grid generation and post-processing of flow solver data were developed

Payload motion control for a varying length flexible gantry crane

Cranes play a very important role in transporting heavy loads in various industries. However, because of its natural swinging characteristics, the control of crane needs to be considered carefully. This paper presents a control approach to a flexible cable crane system in consideration of both rope length varying and system constraints. At first, from Hamilton\u27s extended principle the equations of motion that characterized coupled transverse-transverse motions with varying rope length of the gantry are obtained. The equations of motion consist of a system of partial differential equations. Then, a barrier Lyapunov function is used to derive the control located at the trolley end that can precisely position the gantry payload and minimize vibrations. The designed control is verified through extensive experimental studies

Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft