Model Reduction, Centering, and the Karhunen-Loeve Expansion

Sonja Glavaski, CDS/EE, Caltech

Aeroengine compressor systems have became a subject of major interest to control engineers in recent years, even though the phenomena of stall in compression systems has been studied almost continuously since the early development of the gas turbine engine. To avoid the development of rotating stall--a compressor instability causing a sudden drop in performance--feedback control is necessary, and the most meaningful approach to control design is via low-order models.

The simplest model that adequately describes the basic dynamics of rotating stall is a three state nonlinear model of Moore and Greitzer (MG3) which is a Galerkin truncation onto a first Fourier mode of the full Moore-Greitzer model developed in Moore and Greitzer [1986]. The classical approach to model reduction using Galerkin's method and the Karhunen-Loeve expansion (KLE) attempts to find an approximate solution $\hat{u}(x,t)$ to the solution $u(x,t)$ of a PDE in the form of a truncated series expansion given by $$\hat{u} (x,t) = \sum_{n=1}^N a_n(t)\varphi_n(x),$$ where the mode functions $\varphi_n(x)$ are based on empirical data and are generated by the standard KL methods. For systems with rotational (periodic) symmetry, the mode functions are Fourier modes and the order of the reduced model determined by reasonable criteria for the truncation point is not always small. For a PDE having a traveling wave as a solution, this approach will normally not give satisfactory results.

The disadvantage of making a Galerkin truncation onto a non-propagating function with a fixed spatial shape is that it does not properly describe how the stall cell propagates and evolves in simulations. One usually observes that the stall cell quickly develops a spatial structure having the shape of a square wave with a fixed depth, which then widens until it stabilizes at a fixed width. To capture this behavior with non-propagating modes of fixed spatial shape, one needs to include many modes. A remedy for this is to try to capture the dynamics with a family of propagating curves.

In this paper we propose a new, and computationally efficient modeling method that captures existing translation symmetry in a compression system (and more generally for systems with a rotational symmetry) by finding an approximate solution of the governing PDE in the form of a truncated series expansion of the form $$ \hat{u} (x,t) = \sum_{n=1}^N a_n(t)\varphi_n(x + d(t)). $$ To generate the optimal basis functions $\varphi_n(x)$ prior to performing >KLE, we process the available data set using a ``centering'' procedure which involves giving an appropriate definition of the center of a wave and moving it to a standard position. The eigenvalues of the covariance matrix of ``centered'' data decay rapidly and we obtain a low order approximate system of ODEs.

We have shown that this approach is very efficient in linear and nonlinear scalar wave equations. The method may be viewed as a way of implementing the KLE on the space of solutions of the given PDE modulo a given symmetry group---if one prefers, it is a reduction or orbit space method. Viewed this way, the methodology is quite general and therefore should be useful in a variety of problems. We are currently investigating applying it to a large scale axial compression system dynamics model developed by Mezic [1998].