State-Space Approximations of the Orr-Sommerfeld System with Boundary Inputs and Outputs

Finite-dimensional state-space approximations of the Orr–Sommerfeld equation for plane Poiseuille flow with boundary input and output are discretized to state-space models using two spectral techniques. The models are compared for accuracy and discussed in the context of metrics important for a nonnormal dynamical system. Both state-space models capture the sensitivity behavior of the pole-zero perturbations and the pseudospectrum properties. The effects of balanced truncation ordered by Hankel singular values are discussed. For practical purposes, both methods are effective, and yet certain basic variations in the state-space properties exist. The merits and penalties of the two techniques are also provided. I. Introduction D ISTRIBUTED-PARAMETERS dynamical systems are described by partial differential equations (PDE) with associated boundary conditions. These systems are solved by numerical discretization to yield sets of ordinary differential equations similar to those in lumped-parameters systems [1,2]. This paper considers the state-space modeling of the two-dimensional (2-D) perturbations of plane Poiseuille flow described by the well-known Orr– Sommerfeld (OS) equation with the incorporation of an input/output structure. Two different spectral techniques are evaluated here. Of interest is existence of some basic variation in the state-space properties consisting of certain nontruncatable superdamped eigenvaluesandaninputfeedthroughtermintheoutputequation.Astrong motivation exists todemonstrate the effectiveness of using the finitedimensional state-space approximations in representing an infinitedimensional system. The state-space representation is increasingly popular,becausenumeroustoolboxeshavebeendevelopedbasedon the state-space form. MATLAB®, for example, offers toolboxes for balancing, model reduction, control design, data analysis, Simulink, and system theory [3]. All require standard state-space representation.Foraplanegeometry,spectraltechniques[4,5]arepowerfulin constructing state-space models. These are the original models that represent the infinite-dimensional systems. For certain flows, excitable by the input and observable through the output, the original models can be further balanced and order reduced. The balanced reduced-order models (BROMs) are commonly used in designing estimators and controllers. The OS system is of interest in flow controls, in association with thefundamentalproblemoftransitiontoturbulencein fluid flows ata much lower Reynolds number than the ones predicted by linear stability theory [6]. For the plane Poiseuille flow, the OS differential operators are highly nonnormal [7,8]. It is well known that, for nonnormal systems, the eigenvalues alone are inadequate to predict the stability of the flow. It can be demonstrated that, even when all eigenmodes are damped, large bursts of transient responses (orders of magnitude greater than the excitations) can be generated for certain transfer functions of a nonnormal system. Although exerting no exponential growth, these large transient bursts are believed as capable of triggering a transition to turbulence via a subcritical route to instability [6–8]. The aim of this paper is to compare and validate two spectral techniques applied to the OS system and to point out the basic differences between the eigenvalue properties and the state-space representations. A second aim is to demonstrate the sensitivity of the nonnormality of the OS operator [7] and of the transfer function, as well as to show the effects of balanced model truncation.