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This paper investigates the effect of model uncertainty on the nonlinear dynamics of a generic aeroelastic system. Among the most dangerous phenomena to which these systems are prone, Limit Cycle Oscillations are periodic isolated responses triggered by the nonlinear interactions among elastic deformations, inertial forces, and aerodynamic actions. In a dynamical systems setting, these responses typically emanate from Hopf bifurcation points, and thus a recently proposed framework, which address the problem of robustness from a nonlinear dynamics viewpoint, is employed. Briefly, the notion of robust bifurcation margin extends the concept of μ analysis technique from the robust control theory. The main contribution of this article is a systematic investigation of the numerous scenarios arising in the study of nonlinear flutter when uncertainties in the model are accounted for in the analyses. The advantages of adopting this framework include the possibility to: quantify relevant information for the determination of the nonlinear stability envelope; gain a more in-depth understanding of the physical mechanisms triggering subcritical and supercritical Hopf bifurcations; and reveal properties of the nominal system by identifying isolated branches not straightforward to detect with conventional numerical approaches., Nonlinear Dynamics, 103 (2), ISSN:0924-090X, ISSN:1573-269X

The paper proposes a multi-step identification approach to classify a nonlinear system into qualitatively different regimes and then estimate a low-dimensional subspace where predictions of the original state at future times can be obtained by simulation of low-order dynamics. Proper Orthogonal Decomposition is used to build a library of characteristic modes from training data and is combined with regularization techniques for both the classification and estimation problems. Group Lasso is proposed to more effectively perform the former task. Moreover, l(1) and l(2) regularization problems with singular values weighting of the dynamic modes are suggested to handle the estimation problem in complex scenarios where limited measurement points are available or sensors are noisy. Results obtained on the Rijke tube system, a nonlinear thermoacoustic benchmark problem, demonstrate better classification accuracy and lower prediction error compared with a method from the literature. Copyright (C) 2021 The Authors., IFAC-PapersOnLine, 54 (14), ISSN:2405-8963

This article proposes a framework which allows the study of stability robustness of equilibria of a nonlinear system in the face of parametric uncertainties from the point of view of bifurcation theory. In this context, a branch of equilibria is stable if bifurcations (i.e., qualitative changes of the steady-state solutions) do not occur as one or more bifurcation parameters are varied. The work focuses specifically on Hopf bifurcations, where a stable branch of equilibria meets a branch of periodic solutions. It is of practical interest to evaluate how the presence of uncertain parameters in the system alters the result of analyses performed with respect to a nominal vector field. Note that in this article bifurcation parameters have a different meaning than uncertain parameters. To answer the question, the concept of robust bifurcation margins is proposed based on the idea of describing the uncertain system in a Linear Fractional Transformation fashion. The robust bifurcation margins can be interpreted as nonlinear analogues of the structural singular value, or $\mu$, which provides robust stability margins for linear time invariant systems. Their computation is formulated as a nonlinear program aided by a continuation-based multistart strategy to mitigate the issue of local minima. Application of the framework is demonstrated on two case studies from the power system and aerospace literature., SIAM Journal on Applied Dynamical Systems, 19 (3), ISSN:1536-0040

This paper is concerned with the experimental characterization of a thermoacoustic demonstrator known as the Rijke tube. An approach which leverages the prior qualitative knowledge of the sys-tem's behaviour to plan the experiment campaign and identify reliable models of the system is described. First, nonlinear features of the dynamics, e.g. the type of Hopf bifurcation triggered by the thermoacoustic coupling and the periodic orbits arising from it, are investigated. Then, a family of linear systems, parameterized with a coefficient related to the heat release, is constructed. Different approaches for input design and identification are tested. The results show that the identified models can favourably complement the prior knowledge gained from first principles and provide quantitative assessments on the linear and nonlinear behaviour of the system.

In this article the problem of estimating the Region of Attraction (ROA) for polynomial nonlinear systems subject to modeling uncertainties is studied. Based on recent theoretical studies on the calculation of positively invariant sets, this article proposes an optimization problem which allows robust inner Estimates of the Region of Attraction (rERA) to be evaluated. The uncertainties, which can generically be time-invariant or time-varying, are described as semialgebraic sets, and the problem is solved numerically by means of Sum Of Squares relaxations, which allow set containment conditions to be enforced. The ensuing optimization entails non-convex constraints, and an iterative algorithm to enlarge the provable invariant level set is discussed. The proposed algorithm is applied to two study cases of increasing complexity. Further, in order to benchmark the proposed rERA algorithm, comparisons are shown with a class of well established algorithms based on Lyapunov functions level sets. The results showcase the prowess of the proposed approach and its advantages in terms of accuracy and computational time, particularly as the size of the system increases.

A method is presented to estimate the region of attraction (ROA) of stochastic systems with finite second moment and uncertainty-dependent equilibria. The approach employs Polynomial Chaos (PC) expansions to represent the stochastic system by a higher-dimensional set of deterministic equations. We first show how the equilibrium point of the deterministic formulation provides the stochastic moments of an uncertainty-dependent equilibrium point of the stochastic system. A connection between the boundedness of the moments of the stochastic system and the Lyapunov stability of its PC expansion is then derived. Defining corresponding notions of a ROA for both system representations, we show how this connection can be leveraged to recover an estimate of the ROA of the stochastic system from the ROA of the PC expanded system. Two optimization programs, obtained from sum-of-squares programming techniques, are provided to compute inner estimates of the ROA. The first optimization program uses the Lyapunov stability arguments to return an estimate of the ROA of the PC expansion. Based on this result and user specifications on the moments for the initial conditions, the second one employs the shown connection to provide the corresponding ROA of the stochastic system. The method is demonstrated by two examples. © 2020 Elsevier Ltd, Automatica, 122, ISSN:0005-1098

The paper discusses an extension of μ (or structured singular value), a well-established technique from robust control for the study of linear systems subject to structured uncertainty, to nonlinear polynomial problems. Robustness is a multifaceted concept in the nonlinear context, and in this work the point of view of bifurcation theory is assumed. The latter is concerned with the study of qualitative changes of the steady-state solutions of a nonlinear system, so-called bifurcations. The practical goal motivating the work is to assess the effect of modeling uncertainties on flutter, a dynamic instability prompted by an adverse coupling between aerodynamic, elastic, and inertial forces, when considering the system as nonlinear. Specifically, the onset of flutter in nonlinear systems is generally associated with limit cycle oscillations emanating from a Hopf bifurcation point. Leveraging μ and its complementary modeling paradigm, namely linear fractional transformation, this work proposes an approach to compute margins to the occurrence of Hopf bifurcations for uncertain nonlinear systems. An application to the typical section case study with linear unsteady aerodynamic and hardening nonlinearities in the structural parameters will be presented to demonstrate the applicability of the approach., CEAS Aeronautical Journal, 11 (4)

The Region of Attraction of an equilibrium point is the set of initial conditions whose trajectories converge to it asymptotically. This article, building on a recent work on positively invariant sets, deals with inner estimates of the ROA of polynomial nonlinear dynamics. The problem is solved numerically by means of Sum Of Squares relaxations, which allow set containment conditions to be enforced. Numerical issues related to the ensuing optimization are discussed and strategies to tackle them are proposed. These range from the adoption of different iterative methods to the reduction of the polynomial variables involved in the optimization. The main contribution of the work is an algorithm to perform the ROA calculation for systems subject to modeling uncertainties, and its applicability is showcased with two case studies of increasing complexity. Results, for both nominal and uncertain systems, are compared with a standard algorithm from the literature based on Lyapunov function level sets. They confirm the advantages in adopting the invariant sets approach, and show that as the size of the system and the number of uncertainty increase, the proposed heuristics ameliorate the commented numerical issues., Journal of the Franklin Institute, 356 (8), ISSN:0016-0032, ISSN:1879-2693

This article focuses on the numerical estimation of the Region of Attraction of systems with polynomial vector field. The presented approach, based on a recent theoretical work on positively invariant sets, computes the inner Estimates of the Region of Attraction by means of Sum of Squares techniques. This allows the set containment conditions defining the region to be enforced at the expense of requiring iterative schemes since the ensuing optimization features bilinearities in the decision variables. The main contribution consists of two novel algorithms aimed at addressing some of the shortcomings typically associated with the adoption of iterative schemes. The results confirm the advantages of the proposed approaches, particularly as the size of the system increases.

Dynamic aeroelastic behavior of structurally nonlinear Joined Wings is presented. Three configurations, two characterized by a different location of the joint and one presenting a direct connection between the two wings (SensorCraft-like layout) are investigated. The snap-divergence is studied from a dynamic perspective in order to assess the real response of the configuration. The investigations also focus on the flutter occurrence (critical state) and postcritical phenomena. Limit Cycle Oscillations (LCOs) are observed, possibly followed by a loss of periodicity of the solution as speed is further increased. In some cases, it is also possible to ascertain the presence of period doubling (flip-) bifurcations. Differences between flutter (Hopf

Two approaches to tackle the nonlinear robust stability problem of an aerospace system are compared. The first employs a combination of the describing function method and μ analysis, while the second makes use of integral quadratic constraints (IQCs). The model analyzed consists of an open-loop wing's airfoil subject to free play and linear time-invariant parametric uncertainties. The key steps entailed by the application of the two methodologies and their main features are critically discussed. Emphasis is put on the available insight on the nonlinear postcritical behavior known as limit cycle oscillation. It is proposed a strategy to apply IQCs, typically used to find absolute stability certificates, in this scenario, based on a restricted sector bound condition for the nonlinearity. Another contribution of this paper is to understand how the conservatism usually associated with the IQCs multipliers selection can be overcome by using information coming from the first approach. Nonlinear time domain simulations showcase the prowess of these approaches in estimating qualitative trends and quantitative response's features., IEEE Transactions on Control Systems Technology, 27 (2), ISSN:1063-6536, ISSN:1558-0865

Freeplay is a nonlinearity commonly encountered in aeroservoelastic applications which is known to cause Limit Cycle Oscillations (LCOs), limited amplitude flutter phenomena not captured by a linear analysis. Uncertainties in the models are also known to play an important role in triggering instabilities which might not be present in the nominal case, or altering their features in an unpredictable way. This paper shows the process to build a framework to study the nonlinear behavior of a typical section affected by a freeplay in the control surface and uncertainties in its parameters’ values. Starting from the definition of the nominal aeroelastic model, the nonlinear framework is implemented by means of the Describing Function (DF) method and robust analysis is introduced by means of \(\mu \) technique. In addition, it is shown an idea to perform a tailoring of the LCO graph of the system with the practical goal to limit the oscillation amplitude. Implications and advantages of using DF and \(\mu \) as primary tools are highlighted, and prowess of the methodology is showcased with an example.

Dynamic aeroelastic behaviour of structurally nonlinear Joined Wings is presented. Three configurations, two characterized by a different location of the joint and one presenting a direct connection between the two wings (Sensorcraft-like layout) are investigated. As a first step, the snap-divergence is studied from a dynamic perspective in order to assess the real response of the configuration. Later on, the investigations focus on the flutter occurrence (critical state) and go further in analyzing postcritical phenomena: Limit Cycle Oscillations (LCOs) are observed followed by a lost of periodicity of the solution as speed is further increased. In some cases, it is also possible to ascertain the presence of period doubling (flip-) bifurcation. Differences between flutter (Hopf ’s bifurcation) speed evaluated with linear and nonlinear analyses are discussed in depth, in order to understand if a less computationally intense approach may be used with confidence. Both frequency and time-domain approaches are compared. Moreover, aerodynamic solvers based on the potential flow are critically examined and discussed. In particular, it is assessed in what measure the use of more sophisticated (and computationally more intensive) aerodynamic and interface models impacts the aeroelastic predictions. These differences range from the methods adopted for load transferring to the models employed to describe the wake. When the use of the tools gives different results, a physical interpretation of the leading mechanism generating the mismatch is provided. In particular, for PrandtlPlane-like configurations the aeroelastic response is very sensitive to the wake’s shape. As a consequence, it is suggested that a more sophisticated modelling of the wake positively impacts the reliability of aerodynamic and aeroelastic analysis. For Sensorcraft-like configurations some LCOs are characterized by a non-synchronous motion of the inner and outer portion of the lower wing: the wings’ tip exhibits a small oscillation during the descending or ascending phase, whereas the mid-span station describes a sinusoidal-like trajectory in the time domain.

Two approaches to tackle the nonlinear robust stability problem of a plant are considered. The first employs a combination of the Describing Function method and μ analysis, while the second makes use of Integral Quadratic Constraints (IQCs). The model analyzed consists of an open-loop wing's airfoil subject to freeplay and LTI parametric uncertainties. One of the main contributions of the work is to provide methodologies to quantitatively determine the post-critical behaviour of the system, known as Limit Cycle Oscillation (LCO). When the first approach is adopted, this is studied by means of a worst-case LCO curve, whose definition is given in the paper. The IQC framework, typically used to find asymptotic stability certificates, is applied to this scenario by introducing a restricted sector bound condition for the nonlinearity.

A general framework for Region of Attraction (ROA) analysis is presented. The considered system consists of the feedback interconnection of a plant with polynomial dynamics and a bounded operator. The input/output behavior of the latter is characterized using an Integral Quadratic Constraint (IQC), for which it is assumed an hard factorization holds. This formulation allows to analyze problems involving hard-nonlinearities and uncertainties, adding to the state of practice typically limited to polynomial vector fields. An iterative algorithm based on Sum of Squares optimization is proposed to compute inner estimates of the ROA. The effectiveness of this approach is demonstrated on a numerical example featuring a nonlinear closed-loop system with saturated inputs.

A general framework is presented to estimate the Region of Attraction of attracting equilibrium points. The system is described by a feedback connection of a nonlinear (polynomial) system and a bounded operator. The input/output behavior of the operator is characterized using an Integral Quadratic Constraint. This allows to analyze generic problems including, for example, hard-nonlinearities and different classes of uncertainties, adding to the state of practice in the field which is typically limited to polynomial vector fields. The IQC description is also nonrestrictive, with the main result given for both hard and soft factorizations. Optimization algorithms based on Sum of Squares techniques are then proposed, with the aim to enlarge the inner estimates of the ROA. Numerical examples are provided to show the applicability of the approaches. These include a saturated plant where bounds on the states are exploited to refine the sector description, and a case study with parametric uncertainties for which the conservativeness of the results is reduced by using soft IQCs., Automatica, 109, ISSN:0005-1098