Koopman form of nonlinear systems with inputs

The Koopman framework proposes a linear representation of finite-dimensional nonlinear systems through a generally infinite-dimensional globally linear embedding. Originally, the Koopman formalism has been derived for autonomous systems. In applications for systems with inputs, generally a linear time invariant (LTI) form of the Koopman model is assumed, as it facilitates the use of control techniques such as linear quadratic regulation and model predictive control. However, it can be easily shown that this assumption is insufficient to capture the dynamics of the underlying nonlinear system. Proper theoretical extension for actuated continuous-time systems with a linear or a control-affine input has been worked out only recently, however extensions to discrete-time systems and general continuous-time systems have not been developed yet. In the present paper, we systematically investigate and analytically derive lifted forms under inputs for a rather wide class of nonlinear systems in both continuous and discrete time. We prove that the resulting lifted representations give Koopman models where the state transition is linear, but the input matrix becomes state-dependent (state and input-dependent in the discrete-time case), giving rise to a specially structured linear parameter-varying (LPV) description of the underlying system. We also provide error bounds on how much the dependency of the input matrix contributes to the resulting representation and how well the system behavior can be approximated by an LTI Koopman representation. The introduced theoretical insight greatly helps for performing proper model structure selection in system identification with Koopman models as well as making a proper choice for LTI or LPV techniques for the control of nonlinear systems through the Koopman approach.</p

Identification of input-output LPV models

This chapter presents an overview of the available methods for identifying input-output LPV models both in discrete time and continuous time with the main focus on noise modeling issues. First, a least-squares approach and an instrumental variable method are presented for dealing with LPV-ARX models. Then, a refined instrumental variable approach is discussed to address more sophisticated noise models like Box-Jenkins in the LPV context. This latter approach is also introduced in continuous time and efficient solutions are proposed for both the problem of time-derivative approximation and the issue of continuous-time modeling of the noise

Damage localization in data-driven vibration-based structural health monitoring using linear quadratic estimation theory

Vibration-based Structural Health Monitoring (SHM) is classically approached from two different directions; both involve the acquisition and processing of vibration signals. The first and most popular strategy, which is also followed in the present thesis, relies entirely on the measurements. In contrast, the second approach employs physical models such as finite element (FE) models that are designed based on mechanical principles. In times in which the real-time processing of digital twins for engineering structures becomes more and more realistic, model-based approaches for vibration-based SHM receive increasing attention. Data-driven strategies are still primarily used in vibration-based SHM, and they will remain appealing in situations where precise physical modeling appears cumbersome. Hence, the need for efficient, robust, and reliable data-driven techniques concerning all stages and hurdles of SHM that can prove themselves in practice will never vanish. In this regard, after over 25 years of research, the number of real-life validation studies is still surprisingly low. As for all SHM strategies, the difficulty concerning damage analysis increases with higher levels of realization. Beginning with the goal of detecting damage, SHM finally seeks to predict the remaining lifetime of a structure. The intermediate steps comprise the localization, classification, and assessment of damage. Without the existence of adequately calibrated physics-based models, the successful implementation of methods tackling the objectives beyond damage localization in an unsupervised data-driven scheme is questionable. The term ‘unsupervised’ refers to the fact that knowledge about the manifestation of damage is not available. Especially in civil engineering, this situation pertains in general and is considered in the present thesis. In data-driven SHM, where the area of structural alterations is narrowed down to adjacent sensors, damage localization suffers from the coarse spatial resolution of parsimonious data acquisition systems. Classical modal approaches that hold potential for damage localization require a dense sensor network or significant damage. Originating from the field of fault detection and isolation, estimator- and filter-based methods have proven to be applicable for damage identification of mechanical and civil engineering structures. Notably, they feature an enormous sensitivity towards structural changes when properly designed. Although it remains advantageous for the sake of precise damage localization, these tools such as Kalman or H-infinity filters do not exhibit the inherent demand for a dense sensor network. Consequently, they promise to be viable techniques for the application in vibration-based SHM. A central challenge of this discipline is the discrimination between the natural variability of the structure’s dynamics and the one caused by damage. The former results from varying environmental and operational conditions (EOCs). Especially highly sensitive methods for damage identification are affected by these natural changes, and thus, rely on an efficient data normalization strategy, which can prove itself in practice. In light of these challenges, this thesis provides a real-life validation for the application of quadratic estimators in data-driven vibration-based SHM. To this end, an elaborate technique for estimator-based damage localization is adapted and included in an SHM framework comprising the necessary steps of data normalization and statistical testing. The damage analysis methodology was originally designed for H-infinity filters, which seem well-suited for use in SHM, as they do not assume specific properties of the excitation acting on the structure nor of the involved disturbances. However, previous studies have shown that, in some cases, the filter performance required to achieve high levels of sensitivity towards localized damage cannot be obtained. This issue can be circumvented by employing well-tuned Kalman filters. Therefore, a novel approach for noise covariance estimation is established at first. The associated estimation scheme constitutes a parametric extension of the popular autocovariance least-squares (ALS) technique. The effectiveness of this estimation technique in the context of Kalman filter-based damage localization is studied first using simulations and laboratory experiments. The second part is dedicated to the problem of handling EOCs. This body of work proposes an identification scheme for linear parameter-varying systems based on the interpolation of linear time-invariant systems for different operating points. A simulation study demonstrates the applicability for the purpose of data normalization. Finally, real-life validation of the proposed methods for SHM is conducted. Therefore, a steel lattice mast located outdoors functions as the test object. It is naturally affected by ambient sources of excitation, variability, and uncertainty. The mast, explicitly designed for this validation purpose, is equipped with reversible damage mechanisms that may be activated or removed to reduce the stiffness at multiple locations of the structure. The investigations conducted in this part of the thesis demonstrate proper damage detection of all considered damages as well as localization for the highest degree of severity. These promising results suggest the applicability of the presented methods for Kalman filter tuning, damage localization, and data-normalization in the context of vibration-based SHM

Parameterized macromodeling of passive and active dynamical systems

L'abstract è presente nell'allegato / the abstract is in the attachmen

Estimation in multisensor networked systems with scarce measurements and time varying delays

In this paper, the problem of estimating signals from a dynamic system at regular periods from scarce, delayed and possibly time disordered measurements acquired through a network is addressed. A model based predictor that takes into account the delayed and irregularly gathered measurements from different devices is used. Robustness of the predictor to the time-delays and scarce data availability as well as disturbance and noise attenuation is dealt with via H∞ performance optimization. The result is a time variant estimator gain that depends on the measurement characteristics, but belonging to an offline precalculated finite set, and hence, the online needed computer resources are low. An alternative to reduce the number of gains to be stored has been proposed, based on defining the gain as a function of the sampling parameters. The idea allows reaching a compromise between online computer cost and performance.This work has been granted by the MICINN project number DPI2008-06731-C02-02/DPI.Albertos Pérez, P. (2012). Estimation in multisensor networked systems with scarce measurements and time varying delays. Systems and Control Letters. 61(4):555-562. https://doi.org/10.1016/j.sysconle.2012.02.009S55556261