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Impact of measurement uncertainties on data-based models
In both system identification and data assimilation, the quality of the data plays an important role. As the data quality impacts nearly all design steps, any uncertainty in the data will influence the identified model parameters or the recovered variables, and will ultimately affect the performance and robustness margins of the controller (for system identification) or the accuracy and validity of the reconstructed flow fields (for data assimilation). System identification and data assimilation are par-ticularly attractive for processing experimental or real-life data; but these data are often contaminated by deterministic or stochastic perturbations of unknown origin. It is thus important to address the issue of how data uncertainty propagates through the entire system-identification or data-assimilation process and manifests itself in the respective output models. A common procedure to deal with the influence of noise in the data is filtering. This technique is most effective, if the frequency content and nature of the noise source is known or can be rather accurately estimated. For system identification, the filtering of incoming noise is inherent for under-parameterized models, i.e. models whose order is too low to recover the learning data-set accurately. In this case, the identified model acts as a filter.
Control design based on data-sequences
Convectively dominated flows are characterized by information propagation largely in the downstream direction. Consequently, a control setup, respecting this feature, has to be designed in a feedforward configuration. The goal of flow control efforts is the reduction of disturbance levels measured by downstream sensors y, also referred to as cost sensors. Since the source of these disturbances is assumed to be mainly upstream, actuators u have to be placed upstream of the sensors. To complete the control setup, spy-sensors s will be placed upstream of the actuators. Their role is the detection and estimation of the incoming disturbance environment – information that is valuable for an effective control design u to accomplish our cost objective (measured by y). The resulting configuration of spy-sensors, actuators and cost-sensors is depicted in figure 1. The fluid system is then characterized by two sets of input (the known actuator signals u and the unknown disturbance environment w) and two sets of output (the measure-ments y yielding the control objective and measurements s providing information about the incoming disturbances w). For the sake of simplicity, but without loss of generality, only single input- and output-signals are assumed; a generalization to multiple inputs and outputs will be addressed later. The setup above can be formulated in the terms of transfer functions according to y = Gwyw + Guyu (2.1).
which describes the dependency of the downstream cost measurement y on the distur-bance environment w and the control u. Similarly,the spy-sensor output s is expressed in terms of the true disturbance environment w by writing s = Gwsw. (2.2).
Owing to the convective nature of the flow, no influence of the control u on the spy-sensor s is assumed. Ultimately, we wish to determine a transfer function Csu which links.
Finite-impulse responses (FIR)
To take advantage of a data-based approach, an input- output data sequence of N samples will be recorded from which the transfer functions will be determined. This latter process can be divided into two steps: a model-structure for the system’s impulse responses has to be chosen first, after which a fitting procedure will determine the inherent parameters of the selected model. It is important to choose an input signal u that properly excites the inherent fre-quencies of the system and thus provides a complete input-output map that accurately represents the response behavior of the system to a range of harmonic excitations. To this end, a pseudo-random binary signal (PRBS), a chirp signal or, simply, white noise are appropriate and common choices of a frequency-rich input signal.
FIR model structure
Among the many options to represent a transfer function of a linear system, one of the most straightforward is the finite-impulse response description. For discrete times, this description links the present output to past inputs in the form ∞ y = Guyu⇒ y(k) = Hj u(k − j) (2.7) j=0.
where y(k) is a short-form for y(kΔt) with Δt as the discrete time-step, and Hj stands for the jth impulse response coefficient (also referred to as the jth Markov parameter). Model-predictive control design for convectively unstable flows 5 Under the assumption that, after a sufficient time, the influence of past actuation on the present measurement is negligible, we can truncate the above expression and arrive at the Finite Impulse Response (FIR) model of order µ µ−1 y(k) = Hj u(k − j) (2.8) j=0.
where only the µ first Markov parameters are accounted for. For single input and sin-gle output signals, the Markov parameters are scalar; for nu input signals u and ny measurement signals y the Markov parameters will be ny × nu matrices. For our control configuration (see figure 1) two transfer functions need to be identified: Guy and Gwy G−ws1. Consequently, two sets of Markov parameters, denoted by Hu and Hs, describe the FIR input-output relation,
Table of contents :
1 Introduction
1.1 Classification of flow control methods
1.2 Focus on interactive control
1.3 Model design by system identification
1.4 Model design by data assimilation
1.5 Impact of measurement uncertainties on data-based models
1.6 Outline
2 System identification for flow control
2.1 Introduction
2.2 Article 1: Data-based model-predictive control design for convectively dominated flows
2.3 Article 2: Uncertainty propagation in the design process of data-based flow controllers
3 Experimental flow over an airfoil: reaching the limit of linear identification
3.1 Introduction
3.2 Experimental report
4 Data-assimilation of mean flows
4.1 Introduction
4.2 Article : Data-assimilation for mean flow and shear stress reconstruction
4.3 Article : A data-assimilation method for Reynolds-Averaged Navier-
Stokes-driven mean flow reconstruction
4.4 Article : Data-assimilation of a 2D PIV measurement over an idealized airfoil
5 Conclusions and outlook
5.1 System identification
5.2 Data-assimilation
A Uncertainty propagation within the control design algorithm
A.1 LQG control-design procedure
A.2 Uncertainty propagation through controller design
Bibliography .
