Balanced truncation model reduction of periodic systems

The balanced truncation approach to model reduction is considered for linear discrete-time periodic systems with time-varying dimensions. Stability of the reduced model is proved and a guaranteed additive bound is derived for the approximation error. These results represent generalizations of the corresponding ones for standard discrete-time systems. Two numerically reliable methods to compute reduced order models using the balanced truncation approach are considered. The square-root method and the potentially more accurate balancing-free square-root method belong to the family of methods with guaranteed enhanced computational accuracy. The key numerical computation in both methods is the determination of the Cholesky factors of the periodic Gramian matrices by solving nonnegative periodic Lyapunov equations with time-varying dimensions directly for the Cholesky factors of the solutions

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

Balanced Truncation Model Reduction of Nonlinear Cable-Mass PDE System

We consider model order reduction of a cable-mass system modeled by a one dimensional 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 the left boundary of the cable. A mass-spring model at the right end of the cable includes a nonlinear stiffening force. 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 right boundary. We believe the nonlinear cable-mass model considered here has not been explored elsewhere; therefore, we prove the well-posedness and exponential stability of the unforced linear and nonlinear models 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 understood about model reduction of nonlinear input-output systems. Therefore, 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. We also prove the well-posedness and exponential stability of other cable-mass problems with unbounded input and output operators, and numerically investigate the behavior of the ROMs for these system

Balanced truncation model reduction for semidiscretized Stokes equation

We discuss model reduction of linear continuous-time descriptor systems that arise in the control of semidiscretized Stokes equations. Balanced truncation model reduction methods for descriptor systems are presented. These methods are closely related to the proper and improper controllability and observability Gramians and Hankel singular values of descriptor systems. The Gramians can be computed by solving projected generalized Lyapunov equations. Important properties of the balanced truncation approach are that the asymptotic stability is preserved in the reduced order system and there is a priori bound on the approximation error. We demonstrate the application of balanced truncation model reduction to the semidiscretized Stokes equation