{'fraction_non_alphanumeric': 0.02750809061488673, 'fraction_numerical': 0.0, 'mean_word_length': 5.544973544973545, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Empirical pseudo-balanced model reduction and feedback control of weakly nonlinear convection patterns

An empirical model reduction method is performed on nonlinear transient convection patterns near the threshold, using a pseudo-inverse-based projective method called the pseudo-balanced proper orthogonal decomposition (PBPOD). These transient patterns are large-scale amplitude/phase modulations in convection rolls, obtained by prescribing selected spatial input-shape functions. For the nonlinear convection patterns modelled, PBPOD appears to be very effective. Using the nonlinear front example, PBPOD is compared with other existing methods, such as POD and linearized BPOD. The limitations of the methods are discussed. Using a complex prescribed input, complex disturbances are generated and the outputs of the open-loop responses are compared between the original and the reduced-order models. The agreement is good. A feedback-control study is performed using the nonlinear front example. The controller is built by the pseudo-balanced reduced-order model, with a low-order nonlinear estimator. Closed-loop simulations show that the nonlinear travelling fronts can be effectively damped out by the feedback-control actions.