Reduced-order models for control of fluids using the eigensystem realization algorithm

Abstract

As sensors and flow control actuators become smaller, cheaper, and more pervasive, the use of feedback control to manipulate the details of fluid flows becomes increasingly attractive. One of the challenges is to develop mathematical models that describe the fluid physics relevant to the task at hand, while neglecting irrelevant details of the flow in order to remain computationally tractable. A number of techniques are presently used to develop such reduced-order models, such as proper orthogonal decomposition (POD), and approximate snapshot-based balanced truncation, also known as balanced POD. Each method has its strengths and weaknesses: for instance, POD models can behave unpredictably and perform poorly, but they can be computed directly from experimental data; approximate balanced truncation often produces vastly superior models to POD, but requires data from adjoint simulations, and thus cannot be applied to experimental data.In this paper, we show that using the Eigensystem Realization Algorithm (ERA) [15], one can theoretically obtain exactly the same reduced order models as by balanced POD. Moreover, the models can be obtained directly from experimental data, without the use of adjoint information. The algorithm can also substantially improve computational efficiency when forming reduced-order models from simulation data. If adjoint information is available, then balanced POD has some advantages over ERA: for instance, it produces modes that are useful for multiple purposes, and the method has been generalized to unstable systems. We also present a modified ERA procedure that produces modes without adjoint information, but for this procedure, the resulting models are not balanced, and do not perform as well in examples. We present a detailed comparison of the methods, and illustrate them on an example of the flow past an inclined flat plate at a low Reynolds number.

Riccati-less approach for optimal control and estimation: an application to two-dimensional boundary layers

“…A systematic way to perform such a reduction is to use balanced truncation (Moore 1981); for a high-dimensional system, balanced truncation can be approximated using a snapshot-based algorithm proposed by Rowley (2005). This methodology is equivalent to a system identification method called the eigensystem realization algorithm (ERA) (Juang & Pappa 1985), as shown in Ma, Ahuja & Rowley (2011).…”

Reduced Order Modelling of Aircraft Gust Response for Use in Early Design Stages

“…= system matrices ̃, ̃, C, D = discrete-time system matrices C l , C m = lift and pitching moment coefficients H(t) = continuous-time impulse response matrix H rs (k) = Hankel matrix of size r × s at time level k H k = Markov parameter h 1 = first order kernel of the system k = time level l g = gust length m = number of inputs in a multiple input/output system p = number of outputs in a multiple input/output system t = time U = velocity U, V = square unitary matrices v g (t) = input vector x = state vector ẋ = state vector differentiated with respect to time y = output vector y 1 = non-linear system response to a pulse input of arbitrary y 11 = non-linear system response to a pulse input of twice the magnitude as that of y 1 0 m = square identity matrix of size m × m 1 Ph.D Researcher, Department of Aerospace Engineering, Queens Building, University Walk, Bristol, England, BS8 1TR. 2 Senior Lecturer, Department of Aerospace Engineering.…”

Transition delay in a boundary layer flow using active control

“…We keep the first r states, by considering the first r columns of the matrices U ∈ R m o ×m o , the first r rows of the matrix V ∈ R m c ×m c , and the first r rows and columns of Σ . The resulting matrices U r ∈ R m o ×r , V r ∈ R m c ×r and Σ r ∈ R r×r are used for finding the ROM of the system; indeed, the ROM is identified by the matrices A r ∈ R r×r , B r ∈ R r×(d+m) and C r ∈ R (k+p)×r defined as The equivalence of this procedure with balanced truncation can be shown by directly comparing the relations in A.6 with ROM obtained as projection onto a base of balanced modes (see, for more details, Ma et al 2011). The resulting ROM is in time-discrete form, but it can easily converted in continuous-time form (Glad & Ljung 2001) for running the ROM next to the main DNS simulation.…”

“…the linearized Navier-Stokes equation) and the related adjoint. In a recent application, Ma, Ahuja & Rowley (2011) propose for fluid flow control a system identification algorithm usually referred to as eigensystem realization algorithm (ERA) (see Juang & Pappa 1985); they show the equivalence between approximate balanced truncation and ERA, although using the latter algorithm neither simulations of the adjoint system nor a Galerkin projection onto the balanced mode basis are necessary to obtain low-order models. Once the low-order controller is designed, it can be tested in full direct numerical simulations (DNS) or large eddy simulations (LES).…”

“…We follow the reduce-then-design scheme (see e.g. Anderson & Liu 1989); first, ROMs of the system are obtained by applying the ERA, see Juang & Pappa (1985) and Ma et al (2011). The control design is performed by following the LQG approach, applied on the low-order models; for a derivation of the LQG solution, we refer the reader to Lewis & Syrmos (1995) and Dullerud & Paganini (1999).…”