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A self-excited system is a nonlinear system with the property that a constant input yields a bounded, nonconvergent response. Nonlinear identification of self-excited systems is considered using a Lur'e model structure, where a linear model is connected in feedback with a nonlinear feedback function. To facilitate identification, the nonlinear feedback function is assumed to be continuous and piecewise affine (CPA). The present paper uses least-squares optimization to estimate the coefficients of the linear dynamics and the slope vector of the CPA nonlinearity, as well as mixed-integer optimization to estimate the order of the linear dynamics and the breakpoints of the CPA function. The proposed identification technique requires only output data, and thus no measurement of the constant input is required. This technique is illustrated on a diverse collection of low-dimensional numerical examples as well as data from a gas-turbine combustor. 

Computationally efficient modeling of gas turbine combustion is challenging due to the chaotic multi-scale physics and the complex non-linear interactions between acoustic, hydrodynamic, and chemical processes. A large-eddy simulation (LES) is conducted for the model combustor of Meier et al. (1) using an unstructured mesh finite volume method with turbulent combustion effects modeled using a flamelet-based method. The flow field is validated via comparison to averaged and unsteady high-frequency particle image velocimetry (PIV) fields. A high degree of correlation is noted with the experiment in terms of flow field snapshots and via modal analysis. The dynamics of the precessing vortex core (PVC) is quantitatively characterized using dynamic mode decomposition. The validated FOM dataset is used to construct projection-based ROMs, which aim to reduce the system dimension by projecting the state onto a reduced dimensional linear manifold. The use of a structure-preserving least squares formulation (SP-LSVT) guarantees stability of the ROM, compared to traditional model reduction techniques. The SP-LSVT ROM provides accurate reconstruction of the combustion dynamics within the training region, but faces a significant challenge in future state predictions. This limitation is mainly due to the increased projection error, which in turn is a direct consequence of the highly chaotic nature of the flow field, involving a wide range of disperse coherent structures. Formal projection-based ROMs have not been applied to a problem of this scale and complexity, and achieving accurate and efficient ROMs is a grand challenge problem. Further advances in non-linear manifold projections or adaptive basis projections have the potential to improve the predictive capability of this class of ROMs. 

Description of the DISCo facility and adaptive control experiments 

Koopman decomposition is a nonlinear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman-invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on a multi-task feature learning to extract the most informative Koopman-invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity-promoting extensions of EDMD/KDMD. Furthermore, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship-airwake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analysed, with an emphasis on characterizing transient dynamics. The results are compared with existing theoretical expositions and numerical approximations. 

This paper considers system identification for systems whose output is asymptotically periodic under constant inputs. The model used for system identification is a discretetime Lur’e model consisting of asymptotically stable linear dynamics, a time delay, a washout filter, and a static nonlinear feedback mapping. For sufficiently large scaling of the loop transfer function, these components cause divergence under small signal levels and decay under large signal amplitudes, thus producing an asymptotically oscillatory output. A leastsquares technique is used to estimate the coefficients of the linear model as well as the parameters of a piecewise-linear approximation of the feedback mapping. 

Identification of self excited systems 

In some applications of control, the objective is to optimize the constant asymptotic response of the system by moving the state of the system from one forced equilibrium to another. Since suppression of the transient response is not the main objective, the feedback control law can operate quasistatically, that is, extremely slowly relative to the open-loop dynamics. Although integral control can be used to achieve the desired setpoint, three issues must be addressed, namely, nonlinearity, uncertainty, and multistability, where multistability refers to the fact that multiple locally stable equilibria may exist for the same constant input. In fact, multistability is the mechanism underlying hysteresis. The present paper applies an adaptive digital PID controller to achieve quasi-static control of systems that are nonlinear, uncertain, and multistable. The approach is demonstrated on multistable systems involving unmodeled cubic and backlash nonlinearities. 

Self-excited systems arise in many applications, such as biochemical systems, mechanical systems with fluidstructure interaction, and fuel-driven systems with combustion dynamics. This paper presents a Lur’e model that exhibits biased oscillations under constant inputs. The model involves arbitrary asymptotically stable linear dynamics, time delay, a washout filter, and a saturation nonlinearity. For all sufficiently large scalings of the loop transfer function, these components cause divergence under small signal levels and decay under large signal amplitudes, thus producing an oscillatory response. A bias-generation mechanism is used to specify the mean of the oscillation. The main contribution of the paper is the presentation and analysis of a discrete-time version of this model.