System Gramians Evaluation for Hyperbolic PDEs

Abstract

The paper considers the problem of gramians computation for linear hyperbolic distributed parameter systems. Two different cases are considered: vibrating string and beam systems. The presented approach is based on directly deriving the equations solutions by using time - space separation of variables and the Fourier series representation method. The initial problem framework is based on the state space formulation for infinite dimensional systems. This framework uses Riesz-spectral operators defined over Hilbert spaces and implements the concept of a C0 strongly continuous semigroup generated by bounded system operator. The solution of the hyperbolic partial differential equations is divided in two parts. The zero input part is due to the initial conditions and participates in obtaining the observability gramian of the system. The zero state part is a consequence of the input signal effect and is used to compute the controllability gramian.