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

Empirical Evaluation of the Parallel Distribution Sweeping Framework on Multicore Architectures

In this paper, we perform an empirical evaluation of the Parallel External Memory (PEM) model in the context of geometric problems. In particular, we implement the parallel distribution sweeping framework of Ajwani, Sitchinava and Zeh to solve batched 1-dimensional stabbing max problem. While modern processors consist of sophisticated memory systems (multiple levels of caches, set associativity, TLB, prefetching), we empirically show that algorithms designed in simple models, that focus on minimizing the I/O transfers between shared memory and single level cache, can lead to efficient software on current multicore architectures. Our implementation exhibits significantly fewer accesses to slow DRAM and, therefore, outperforms traditional approaches based on plane sweep and two-way divide and conquer.Comment: Longer version of ESA'13 pape

Feedback Control of Low Dimensional Models of Transition to Turbulence

The problem of controlling or delaying transition to turbulence in shear flows has been the subject of numerous papers over the past twenty years. This period has seen the development of several low dimensional models for parallel shear flows in an attempt to explain the failure of classical linear hydrodynamic stability theory to correctly predict transition. In recent years, ideas from robust control theory have been employed to attack this problem. In this paper we use these models to develop a scenario for transition that employs both classical bifurcation theory and robust control theory. In addition, we present numerical results to illustrate the ideas and to show how feedback can be used to delay transition. We close with a specific conjecture and discuss some previous results along this line

Analysis of Transient Growth in Iterative Learning Control Using Pseudospectra

In this paper we examine the problem of transient growth in Iterative Learning Co ntrol (ILC). Transient growth is generally avoided in design by using robust monotonic convergence (RMC) criteria. However, RMC leads to fundamental performance limitations. We consider the possibility of allowing safe transient growth in ILC algorithms as a means to circumvent these limitations. Here the pseudospectra is used for the first time to study transient growth in ILC. Basic properties of the pseudospectra that are relevant to the ILC problem are presented. Two ILC design problems are considered and examined using pseduospectra. The pseudospectra provides new results for these problems and illuminates the oft-misunderstood problem of transient growth

Balanced POD Algorithm for Robust Control Design for Linear Distributed Parameter Systems

A mathematical model of a physical system is never perfect; therefore, robust control laws are necessary for guaranteed stabilization of the nominal model and also nearby systems, including hopefully the actual physical system. We consider the computation of a robust control law for large-scale finite dimensional linear systems and a class of linear distributed parameter systems. The controller is robust with respect to left coprime factor perturbations of the nominal system. We present an algorithm based on balanced proper orthogonal decomposition to compute the nonstandard features of this robust control law. Numerical results are presented for a convection diffusion partial differential equation

Balanced Proper Orthogonal Decomposition for Model Reduction of Infinite Dimensional Linear Systems

In this paper, we extend a method for reduced order model derivation for finite dimensional systems developed by Rowley to infinite dimensional systems. The method is related to standard balanced truncation, but includes aspects of the proper orthogonal decomposition in its computational approach. The method is also applicable to nonlinear systems. The method is applied to a convection diffusion equation

Balanced POD for Linear PDE Robust Control Computations

A mathematical model of a physical system is never perfect; therefore, robust control laws are necessary for guaranteed stabilization of the nominal model and also nearby systems, including hopefully the actual physical system. We consider the computation of a robust control law for large-scale nite dimensional linear systems and a class of linear distributed parameter systems. The controller is robust with respect to left coprime factor perturbations of the nominal system. We present an algorithm based on balanced proper orthogonal decomposition to compute the nonstandard features of this robust control law. Convergence theory is given, and numerical results are presented for two partial di erential equation systems

Towards Transient Growth Analysis and Design in Iterative Learning Control

In this article the problem of bounding transient growth in iterative learning control (ILC) is examined. While transient growth is not a desirable property, the alternative, robust monotonic convergence, leads to fundamental performance limitations. to circumvent these limitations, this article considers the possibility that some transient growth, if properly limited, is a viable and practical option. Towards this end, this article proposes tools for analysing worst-case transient growth in ILC. the proposed tools are based on pseudospectra analysis, which is extended to apply to ILC of uncertain systems. Two practical problems in norm-optimal ILC weighting parameter design are considered. Using the presented tools, it is demonstrated that successful design in the transient growth regime is possible, i.e. the transient growth is kept small while significantly improving asymptotic performance, despite model uncertainty

A Proper Orthogonal Decomposition Approach to Approximate Balanced Truncation of Infinite Dimensional Linear Systems

We extend a method for approximate balanced reduced order model derivation for finite dimensional linear systems developed by Rowley (Int. J. Bifur. Chaos Appl. Sci. Eng. 15(3) (2005), pp. 997-1013) to infinite dimensional systems. The algorithm is related to standard balanced truncation, but includes aspects of the proper orthogonal decomposition in its computational approach. The method can be also applied to nonlinear systems. Numerical results are presented for a convection diffusion system