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Fast triple-parameter extremum seeking exemplified for jet control

Abstract

A fast triple-parameter extremum seeking method is applied for jet control based on the pioneering work of Gelbert et al. (J Process Control 22(4):700, 2012). The simultaneous adaptation of three input parameters takes less time than the single-input adaptation of each parameter combined. The key enablers are phase-shifted sinusoids for the input each of which is evaluated by an extended Kalman filter (EKF). An acceleration of the adaption is obtained by a combined EKF coupling the output to all inputs. The method is illustrated for an analytical optimization problem and experimentally demonstrated for a turbulent jet mixing control. The considered Reynolds numbers \({\hbox {Re}}_D\) based on the jet exit diameter and velocity are 5700, 8000 and 13,300. The main jet is manipulated by a pulsed radially injected minijet which is varied by a mass flow controller and an electromagnetic valve up to high frequencies. The mixing performance is characterized by the centerline jet decay rate and monitored by a hot-wire sensor five diameters downstream at the end of the potential core. The proposed triple-parameter extremum seeking method optimizes the actuation mass flow ratio, frequency and duty cycle. The decay rate increases 11-fold from the unforced reference value of 0.05 to the optimal actuation level of 0.56. The reproducibility is demonstrated with various initial actuation parameters. Moreover, the adaptive control robustly tracks the optimal open-loop actuation for varying \({\hbox {Re}}_D\); the optimal decay rate remains unchanged given the mass flow rate, frequency and duty cycle are optimized. The unforced and actuated flow are investigated with hot wires and visualizations. The three-input ES significantly outperforms a two-parameter optimization for the same configuration in multiple respects (Wu et al. in AIAA J 56(4):1463, 2018): First, the jet decay rate is \(8\%\) faster. Second, the convergence time for three parameters is only \(25 \%\) of the adaptation period of two parameters when \({\hbox {Re}}_D\) is varied. Finally, the current steady-state error is \(45\%\) less than that of the two-parameter optimization. We expect the proposed triple-parameter extremum seeking to be applicable for a large range of flow control experiments.

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Acknowledgements

YZ wishes to acknowledge support given to him from NSFC through Grants 11632006, 91752109 ,91952204 and U1613226. This work is supported by the French National Research Agency (ANR) via the grants ‘FlowCon.’

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A: Parameters of the extended Kalman filter

A: Parameters of the extended Kalman filter

The triple-parameter system is operated with a sampling time of \(\varDelta t = 0.001\) for the simulation and 0.4 s for experiment.

The extended Kalman filter (EKF) is based on estimated covariance matrices of the process and measurement noise. For the analytical examples (Sect. 2), we have used separate EKFs (\(EKF_1\), \(EKF_2\) and \(EKF_3\)) characterized by

$$\begin{aligned} \varvec{Q}= & {} {\varDelta t_n \left[ \begin{array}{cc} 1 &{} \quad 0 \\ 0 &{} \quad 1 \end{array} \right] } \end{aligned}$$
(4)
$$\begin{aligned} \varvec{R}= & {} { k \left[ \begin{array}{cc} 1 &{} \quad 0 \\ 0 &{} \quad 1 \end{array} \right] } \quad \hbox {with}\ k = 0.001. \end{aligned}$$
(5)

The combined EKF is parameterized by

$$\begin{aligned} \varvec{Q}= & {} { \varDelta t_n \left[ \begin{array}{cccc} 1 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 1 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \end{array} \right] } \end{aligned}$$
(6)
$$\begin{aligned} \varvec{R}= & {} { k \left[ \begin{array}{cccc} 1 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 1 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \end{array} \right] }\quad \hbox {with}\ k = 0.001. \end{aligned}$$
(7)

For the jet mixing optimization (Sect. 4), the computation of the combined EKF follows Wu et al. (2018). Different combinations of \(\varvec{Q}\) and \(\varvec{R}\) are evaluated experimentally. The gradients \(\partial K/ \partial C_m\) and \(\partial K/ \partial \alpha \) exceed \(\partial K/ \partial f_a\) by around 40 and 10 times, respectively. Therefore, the corresponding covariance matrix and the process noise is given by

$$\begin{aligned} \varvec{Q}={ \left[ \begin{array}{cccc} 1 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 1 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 40^2 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 10^2 \end{array} \right] } \end{aligned}$$
(8)

and

$$\begin{aligned} \varvec{R}={ \left[ \begin{array}{cccc} 0.01 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0.01 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0.01 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0.01 \end{array} \right] }, \end{aligned}$$
(9)

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Fan, D.W., Zhou, Y. & Noack, B.R. Fast triple-parameter extremum seeking exemplified for jet control. Exp Fluids 61, 152 (2020). https://doi.org/10.1007/s00348-020-02953-3

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