Experimental Tracking of Limit-Point Bifurcations and Backbone Curves Using Control-Based Continuation

Renson, Ludovic; Barton, David A. W.; Neild, Simon A

doi:10.1142/s0218127417300026

Cited by 41 publications

(56 citation statements)

References 46 publications

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“…In this paper, a simplified CBC method that dispenses with numerical continuation algorithms is exploited. The method is briefly explained here and the reader is referred to [20,26] for additional details about the method.…”

Section: 1 Amplitude Sweepsmentioning

confidence: 99%

“…These higher-harmonics cannot be directly removed using the higher-harmonic content of the control signal, i.e.Ū , without altering the dynamics of the underlying uncontrolled experiment or including their amplitudes as additional bifurcation parameters. However, the higher harmonics in the force can be compensate for by adding an external excitation signal that introduces higher harmonics at the input of the experiment [26,32]. This harmonic-compensation procedure was not performed here.…”

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confidence: 99%

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Application of control-based continuation to a nonlinear structure with harmonically coupled modes

Renson

Shaw

Barton

et al. 2019

Mechanical Systems and Signal Processing

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This paper presents a systematic method for exploring the nonlinear dynamics of multi-degree-of-freedom (MDOF) physical experiments. To illustrate the power of this method, known as control-based continuation (CBC), it is applied to a nonlinear beam structure that exhibits a strong 3:1 modal coupling between its first two bending modes. CBC is able to extract a range of dynamical features, including an isola, directly from the experiment without recourse to model fitting or other indirect data-processing methods.Previously, CBC has only been applied to (essentially) single-degree-of-freedom experiments; in this paper we show that the required feedback-control methods and pathfollowing techniques can equally be applied to MDOF systems. A low-level broadband excitation is initially applied to the experiment to obtain the requisite information for controller design and, subsequently, the physical experiment is treated as a black box that is probed using CBC. The invasiveness of the controller used is analysed and experimental results are validated with open-loop measurements. Good agreement between open-and closed-loop results is achieved, though it is found that care needs to be taken in dealing with the presence of higher-harmonics in the force applied to the structure. when compared with linear systems which can be characterised very efficiently with broadband modal analysis techniques. The presence, in nonlinear systems, of multiple coexisting steadystate responses to identical inputs requires repeated tests to capture the possible behaviours. Determining, precisely and with confidence, the regions where these multiple responses exist further multiplies the testing effort as experimental errors such as noise can lead to untimely transitions between these responses even before the region boundaries are reached.Nonlinear systems can also exhibit a wide variety of other complex phenomena not seen in linear systems such as modal interactions [5], quasi-periodic oscillations [6] and isola [7]. The latter correspond to branches of stable responses that are isolated from the main peak of the nonlinear frequency response of the system. Isola can arise through a number of mechanisms such as sub-and super-harmonic resonances on simple single-degree-of-freedom (SDOF) nonlinear systems [6,8,9] as well as through modal interactions. As such, isola have been observed across a wide range of nonlinear mechanical systems, for instance, suspended cables [10], pedestrian bridges [11], vibration absorbers [12,13] and a satellite structure [14]. The reader is referred to [15] for a more detailed review on this topic.The aforementioned challenges associated with nonlinear MDOF structures mean that a robust systematic method for experimental testing is highly desirable. This paper explores the use of control-based continuation (CBC) for this purpose. CBC exploits feedback control to provide a number of advantages compared with classical open-loop approaches to experimental testing. The controller maintains the experiment around a pres...

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Section: 1 Amplitude Sweepsmentioning

confidence: 99%

mentioning

confidence: 99%

Application of control-based continuation to a nonlinear structure with harmonically coupled modes

Renson

Shaw

Barton

et al. 2019

Mechanical Systems and Signal Processing

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“…If no other external excitation is applied to the experiment, the total harmonic excitation applied to the experiment is simply given by Γ = A u 2 1 + B u 2 1 . The reader is referred to [19,20] for a more detailed discussion on this approach that is often referred to as the simplified CBC method.…”

Section: Extracting Periodic Responsesmentioning

confidence: 99%

Numerical continuation in nonlinear experiments using local Gaussian process regression

et al. 2019

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Control-based continuation (CBC) is a general and systematic method to probe the dynamics of nonlinear experiments. In this paper, CBC is combined with a novel continuation algorithm that is robust to experimental noise and enables the tracking of geometric features of the response surface such as folds. The method uses Gaussian process regression to create a local model of the response surface on which standard numerical continuation algorithms can be applied. The local model evolves as continuation explores the experimental parameter space, exploiting previously captured data to actively select the next data points to collect such that they maximise the potential information gain about the feature of interest. The method is demonstrated experimentally on a nonlinear structure featuring harmonically-coupled modes. Fold points present in the response surface of the system are followed and reveal the presence of an isola, i.e. a branch of periodic responses detached from the main resonance peak.

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“…This result will also be exploited in the study of the pluripotency GRN in Section 2.3. Similar principles were also used to capture the periodic orbits and fold bifurcations of several mechanical systems [12][13][14]21].…”

Section: Principles Of Control-based Continuationmentioning

confidence: 99%

“…electro-mechanical) systems: for instance, a parametrically excited pendulum [10] and a bilinear oscillator [11]. We recently used CBC on a multi-degreeof-freedom structure exhibiting complex nonlinear phenomena such as mode interaction, quasi-periodic oscillations and isolated response curves [12,13].…”

Section: Introductionmentioning

confidence: 99%

Exploring the Dynamics of Nonlinear Biochemical Systems using Control-Based Continuation

Gomes

Cesare

Guarino

et al. 2019

Preprint

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Mathematical modelling is routinely used in Systems Biology to understand the mechanisms causing nonlinear phenomena in gene expression, such as switch-like behaviours and temporal oscillations. The reliability of model predictions and bifurcation analysis depend on modelling assumptions and specific choices of model parameters; however, the identification of models is highly challenging due to the complexity of biochemical interactions and noise in experimental data. This paper numerically investigates the use of control-based continuation (CBC) for tracking dynamical features of biochemical systems and, in particular, the bistable dynamics of a gene regulating pluripotency in embryonic stem cells. CBC is a method that exploits feedback control and path following algorithms to explore the dynamic features of a nonlinear physical system directly during experimental tests. CBC applications have so far been limited to non-living (i.e. electro-mechanical) systems. Our numerical simulations show that, in principle, CBC could also be applied to biological experiments to characterise the switch-like dynamics of genes that are important for cell decision making.

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Experimental Tracking of Limit-Point Bifurcations and Backbone Curves Using Control-Based Continuation

Cited by 41 publications

References 46 publications

Application of control-based continuation to a nonlinear structure with harmonically coupled modes

Application of control-based continuation to a nonlinear structure with harmonically coupled modes

Numerical continuation in nonlinear experiments using local Gaussian process regression

Exploring the Dynamics of Nonlinear Biochemical Systems using Control-Based Continuation

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