Model Reduction for Active Control Design Using Multiple-Point Arnoldi Methods

Abstract

I. ABSTRACT A multiple-point Arnoldi method is derived for model reduction of computational fluid dynamic systems. By choosing the number of frequency interpolation points and the number of Arnoldi vectors at each frequency point, the user can select the accuracy and range of validity of the resulting reduced-order model while balancing computational expense. The multiplepoint Arnoldi approach is combined with a singular value decomposition approach similar to that used in the proper orthogonal decomposition method. This additional processing of the basis allows a further reduction in the number of states to be obtained, while retaining a significant computational cost advantage over the proper orthogonal decomposition. Results are presented for a supersonic diffuser subject to mass flow bleed at the wall and perturbations in the incoming flow. The resulting reduced-order models capture the required dynamics accurately while providing a significant reduction in the number of states. The reduced-order models are used to generate transfer function data, which are then used to design a simple feedforward controller. The controller is shown to work effectively at maintaining the average diffuser throat Mach number. II. INTRODUCTIONComputational fluid dynamics (CFD) has reached a considerable level of maturity and is now routinely used in many applications for both external and internal flows. Euler and NavierStokes solvers enjoy widespread use for aerodynamic design and analysis, and provide accurate answers for a variety of complex flows. However, despite ever increasing computational power, unsteady problems are computationally very expensive and time-consuming. More efficient methods for time-varying flow can be obtained if the disturbances are small, and the unsteady solution can be considered to be a small perturbation about a steady-state flow [1]. In this case, a set of linearized equations is obtained which can be time-marched to obtain the flow solution at each instant.Even under the linearization assumption, any CFD-based technique will generate models with a prohibitively high number of states. For this reason, CFD models are not appropriate for many applications where model size and cost are issues. For example, when the aerodynamic solver must be coupled to another disciplinary model, as in aeroelastic analysis or multidisciplinary optimization, CFD models cannot be used. Another application which requires low-order models is control design.The concept of using active control to enhance the stability properties of an unsteady flow has been addressed for several applications [2], [3]. In order to derive control models that will be effective, it is vital that the relevant unsteady flow dynamics are captured accurately. A model is required that will capture not only the dynamics of the disturbance to be controlled, but also the visibility offered by the sensing and the effect on the flow of the actuation mechanism. A high-fidelity CFD code can offer the degree of flow resolution that is required; however, ...

Recent Advances in Structure-Preserving Model Order Reduction

“…see [26,27]. One obvious disadvantage of this approach is that the dimension of the resulting reduced-order model is doubled to 2n.…”

Quasi 1D Modeling of Mixed Compression Supersonic Inlets

“…Primarily, inlet modeling for controls applications has originated from the discipline Computational Fluid Dynamics (CFD) based on discretization of the 3D and 2D Euler and Navier-Stokes equations. 1,2,3 These models have been reduced by developing an orthogonal basis to match selected frequency points or by singular value decomposition 4 to develop reduced order linear models. However, these laborious tasks are not necessary when the same objective can be accomplished starting with 1D CFD models.…”

Parallelization of the Rational Arnoldi Algorithm

“…Our second example is the INLET problem from the Oberwolfach Model Reduction Benchmark Collection [1], an active control model of a supersonic engine inlet; see also [22]. There are two nonsymmetric matrices {A, E} ⊂ R N ×N , a block of vectors B ∈ R N ×2 , and a row vector c T ∈ R 1×N in this problem, where N = 11730.…”

“…8.5]), these methods have seen an increasing number of other applications over the last two decades or so. Examples of rational Krylov applications can be found in model order reduction [11,16,17,22], matrix function approximation [10,13,19], matrix equations [3,9,24], nonlinear eigenvalue problems [20,21,31], and nonlinear rational least squares fitting [5,6].At the core of most rational Krylov applications is the rational Arnoldi algorithm, which is a Gram-Schmidt procedure for generating an orthonormal basis of a rational Krylov space. Given a matrix A ∈ C N,N , a vector b ∈ C N , and a polynomial q m of degree at most m and such that q m (A) is nonsingular, the rational Krylov space of order m is defined as…”