Obtaining impulse response data

The fact that both ERA and BPOD are based exclusively on impulse response information may cause practical difficulties. As mentioned in [228], significant er- rors can result from just measuring the response of a noisy system to an impulse. Apart from ensemble averaging a number of experiments/simulations to reduce the influence of noise, a procedure called the Observer/ Kalman filter identification (OKID) [124] outputs the system Markov parameters given general input-output data. A promising direction for future work would therefore be to study to what extent OKID can be used to obtain balanced models of noisy unstable systems.

In experiments, an additional issue is that it might be difficult to initialise a flow exactly from an unstable equilibrium point if no stabilising controller is known a priori. This point is discussed further in section 4.5.

Projection-free balanced models of

unstable systems: Application to a

one-dimensional system

In section 2.4, we noted that differences can be expected between the theoretical

predictions of chapter 2 and the behaviour of the algorithms in practice. It is

therefore crucial to check to what extent the theoretical conclusions hold in order to show that the projection-free balanced truncation approach can really be used to obtain accurate models for realistic systems in practice. This is the main objective of this chapter, where we apply the method to the linearised complex Ginzburg- Landau equation and analyse its ability to produce accurate reduced-order models. The results presented in this chapter were published in a recent Physical Review E article [75] and presented at the 67th Annual Meeting of the American Physical Society’s Division of Fluid Dynamics.

3.1

Overview of the linearised complex Ginzburg-

Landau equation and simulation setup

The Ginzburg-Landau equation is often used to model convective flows, fluid in- stabilities, and flow control techniques. Reviews on these topics such as [109, 48, 17, 138] frequently demonstrate important concepts with this spatially developing one- dimensional model. It is a common test case for flow control and model reduction studies of convectively and globally unstable flows because its behaviour is often rep- resentative of the Navier-Stokes equations (e.g. [165, 137, 50]), but simulating the

evolution of this system comes at a low computational cost. For more details about the Ginzburg-Landau equation and related studies, the reader is referred to [17].

The linearised complex Ginzburg-Landau equations (3.1a) and the corresponding adjoint equations (3.1b) are:

˙x = Ax =  −ν ∂ ∂χ + γ ∂2 ∂χ2 + µ(χ)  x, (3.1a) ˙x+ = A+x+ =  ν∗ ∂ ∂χ + γ ∗ ∂2 ∂χ2 + µ(χ) ∗  x+, (3.1b)

where χ is the spatial variable, x is the system state, and µ(χ) = µ0− c2u+ µ2χ2/2.

The parameters ν and γ quantify the importance of convection and diffusion in

the system respectively. The growth of instabilities is measured by µ, where µ0 is

analogous to the Reynolds number in the Navier-Stokes equations (it can be used

to change the global stability of the system) and µ2 modifies the parallelism and

normality of the “flow”: a large µ2 corresponds to a large degree of non-parallelism,

while a small value of µ2 and a large value of ν result in a strongly non-normal flow.

Finally the most unstable wavenumber in the flow field is given by cu. Table 3.1

shows the values used here for all these parameters.

The results in this section are based on the matlab code developed by Bagheri et al. [17] with a similar set of parameters to the supercritical (globally unstable) system considered in the article. The main exception is µ0, which was set to 0.57 in

order to obtain a two-dimensional antistable subspace and demonstrate the ability of the projection-free BPOD algorithm to handle several unstable modes. The forward

solver uses a spectral Hermite collocation method: in the discrete problem, the nx

dimensional state is x(χi) for 1 ≤ i ≤ nx. The collocation points χi are the roots of

the nxth Hermite polynomial Hnx(bχ) and the parameter b is chosen to obtain an

accurate approximation of the continuous problem, defined in equation (3.1a). The adjoint equations are based on the discretised forward equations, using the inner-

product x†Qx. The matrix Q is defined so that the energy of the discretised state

approximates the energy of the continuous state. As a result, the adjoint system’s realisation is of the form (Q−1A†Q, Q−1_C†_{, B}†_{Q), as opposed to simply (A}†_{, C}†_{, B}†_).

Further details regarding the discretisation of this code are included in the appendix of [17].

In the present work, the system was set up as a single-input-single-output (SISO) system. Both the input and output have the same narrow Gaussian spatial distri-

bution. The actuator is centred at χI, the upstream limit of the region where

Parameter nx µ0 µ2 ν γ χI, χII cu

Value 220 0.57 −0.01 (2 + 0.2i) (1 − i) ±10.7 0.2

Table 3.1: Ginzburg-Landau equation simulation parameters

at χII, the downstream limit of the unstable region (“branch II” in [17]). The dis-

cretised system has 220 states, corresponding to a spatial extent of [−85, 85], as in [17].

This one-dimensional system was chosen as a first step to keep the cost of simu- lations low. As a result, Gaussian quadrature was not required, and instead a Boole rule quadrature scheme was used. The number of snapshots used N was chosen such

that t∞/N ≈ 0.05. The balancing transformations and ROM were then obtained as

described in section 2.2. As there are only 220 states in the full system, it was also possible to compute the transformations and ROMs based on the various procedures described in sections 2.1.3 and 2.1.4. For brevity, we only compare the projection- free method to the snapshot-based projection method proposed by [20, 5] and the “exact” analogous approach from [66], where the stable subsystem is balanced ex- actly by evaluating the Gramians using the Lyapunov equations (2.6).

Our analysis here is split into two parts. First we investigate the extent to which the key steps of the BPOD algorithm yield the results predicted by theory in chapter 2. We then now turn our attention to the physical problem of the per- formance of reduced-order models obtained with the projection-free algorithm, and compare them with ROMs obtained with existing methods, as described above.