Design and experiment of controlled bistable vortex induced vibration energy harvesting systems operating in chaotic regions

Mechanical Systems and Signal Processing 98 (2018) 1097–1115

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Design and experiment of controlled bistable vortex induced vibration energy harvesting systems operating in chaotic regions B.H. Huynh a, T. Tjahjowidodo b,⇑, Z.-W. Zhong b, Y. Wang c, N. Srikanth d a

Energy Research Institute at NTU, Interdisciplinary Graduate School, Nanyang Technological University, Singapore 639798, Singapore School of Mechanical & Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore c School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore d Energy Research Institute at NTU, Nanyang Technological University, Singapore 639798, Singapore b

a r t i c l e

i n f o

Article history: Received 24 November 2016 Received in revised form 25 April 2017 Accepted 4 June 2017

Keywords: Chaotic responses Bistable spring Vortex induced vibrations (VIV) Energy harvesting Wake oscillator model Poincaré map OGY controller

a b s t r a c t Vortex induced vibration based energy harvesting systems have gained interests in these recent years due to its potential as a low water current energy source. However, the effectiveness of the system is limited only at a certain water current due to the resonance principle that governs the concept. In order to extend the working range, a bistable spring to support the structure is introduced on the system. The improvement on the performance is essentially dependent on the bistable gap as one of the main parameters of the nonlinear spring. A sufficiently large bistable gap will result in a significant performance improvement. Unfortunately, a large bistable gap might also increase a chance of chaotic responses, which in turn will result in diminutive harvested power. To mitigate the problem, an appropriate control structure is required to stabilize the chaotic vibrations of a VIV energy converter with the bistable supporting structure. Based on the nature of the double-well potential energy in a bistable spring, the ideal control structure will attempt to drive the responses to inter-well periodic vibrations in order to maximize the harvested power. In this paper, the OGY control algorithm is designed and implemented to the system. The control strategy is selected since it requires only a small perturbation in a structural parameter to execute the control effort, thus, minimum power is needed to drive the control input. Facilitated by a wake oscillator model, the bistable VIV system is modelled as a 4-dimensional autonomous continuous-time dynamical system. To implement the controller strategy, the system is discretized at a period estimated from the subspace hyperplane intersecting to the chaotic trajectory, whereas the fixed points that correspond to the desired periodic orbits are estimated by the recurrence method. Simultaneously, the Jacobian and sensitivity matrices are estimated by the least square regression method. Based on the defined fixed point and the linearized model, the control gain matrix is calculated using the pole placement technique. The results show that the OGY controller is capable of stabilizing the chaotic responses by driving them to the desired inter-well period-one periodic vibrations and it is also shown that the harvested power is successfully improved. For validation purpose, a real-time experiment was carried out on a computer-based forced-feedback testing platform to validate the

⇑ Corresponding author. E-mail address: [email protected] (T. Tjahjowidodo). http://dx.doi.org/10.1016/j.ymssp.2017.06.002 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

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applicability of the controller in real-time applications. The experimental results confirm the feasibility of the controller to stabilize the responses. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Vortex induced vibration (VIV) is a fluid-structure interaction phenomenon that can be observed in many engineering structures undergoing fluid flows, e.g. risers, pipes, chimneys, suspended cables, mooring lines, etc. In some cases, VIV is an undesired phenomenon when the induced tremendous kinetic energy may result in disastrous fatigue damage of structures. The collapse of Tacoma Narrows Bridge in 1940 is one of the most notorious examples of the incidents. Nevertheless, it has also been realized that plentiful source of energy can be harvested and converted into other utilizable forms of energy from a structure undergoing the phenomenon. The first energy converter based on VIV from water flows was introduced in 2008 and trademarked as VIVACE [1]. The working principle of this energy converter is rather straightforward. The main component of the converter in a form of circular cylinder is transversely immersed into a water flow. The cylinder is supported by elastic springs, where its motions are constrained in one degree-of-freedom, i.e. translation in the cross-flow direction. When the water flow crosses over the cylinder, due to the flow separation, water vortices are formed and alternately shed into two sides of the wake region. The vortex shedding consequently causes periodic drag and lift forces on the cylinder surface. Since the cylinder is constrained in the cross-flow direction, it will vibrate under the effect of the lift force. Connecting the structure to a transmission mechanism to convert the kinetic energy to electrical energy allows us to utilize the harvested energy. The ability to operate efficiently in low velocity water flows is a prominent advantage of the VIV energy converter whereas other types of energy converters, e.g. turbines and watermills, require flow velocities higher than 2 m/s for efficient operations [1]. The induced fluid force excites the VIV structure at the vicinity of its resonance frequency, where the frequency of excitation corresponds to the water flow velocity. Therefore, if the VIV structure is designed with a natural frequency that corresponds to the flow characteristic, it can operate effectively, even in low water flows. The relations of the induced fluid forces, flow velocities and resonance range have been comprehensively studied and reviewed in literature (see e.g. [2–4]). Unfortunately, velocities of natural flows, e.g. ocean and river, might fluctuate. For example, based on a flow measurement at the marine test bed facility of Energy Research Institute at Nanyang Technological University located at the Sentosa Boardwalk, Singapore, the flow velocities fluctuate in the range of 0.1–1.2 m/s in day duration [5]. Meanwhile, the VIV energy converter operates efficiently only at a limited flow range depending on the designed parameters, i.e. the support stiffness, effective mass, damping and length of the cylinder. The variation of the water flow velocity beyond the resonance range will result in a poor performance of the converter since considerable amount of vibrational energy cannot be maintained. This adversity is the most challenging barrier for the practical application of VIV energy converters. Since this issue was revealed, several theoretical and experimental studies have been carried out to improve the performance of a VIV energy converter. They focused on investigating effects of designing parameters including mass, damping, stiffness and surface roughness of the cylinder to improve the vibrating amplitude and broaden the resonance range. A comprehensive review on these studies can be found in [6]. In particular, an apparent benefit from embedding a nonlinear stiffness to a VIV system has been demonstrated in several studies. Experimental studies have been conducted to prove that a hardening stiffness factor has the ability to broaden the resonance range of a VIV energy converter towards the side of high velocity flows [7–9], while theoretical and experimental analyses in [10,11] have indicated that a VIV energy converter enhanced by a bistable spring will improve the resonance range at low velocity flows. The improvement of the performance in the latter case is resulted from the vibrations of the system in the inter-well mode of a bistable oscillator, which in turn allows for larger vibrating amplitude than that with a linear spring. However, it is also shown in these studies that chaotic vibrations (see e.g. [12,13] for more discussions on chaotic vibrations) may occur and lead to a significant drop in the utilizable power at various water flows and diverse ranges of structural parameters, i.e. structural damping, effective mass and bistable gap. In some cases, when a bistable system exhibits chaotic behavior, its utilizable power can drop below 40% of that in the case with a linear spring. The theoretical and experimental analyses in [10] provide detailed study on the bifurcation of a bistable VIV energy converter from periodic to chaotic responses and vice versa, as well as the evolution of chaotic degree as a function of the bistable gap and structural damping. According to this information, chaotic responses can be avoided when the bistable VIV converter is designed with a small bistable gap, i.e. smaller than the value of 0.05D, where D is the diameter of the cylinder. However, the desired effect in improving the utilizable power from the bistable spring is becoming minor in a case of a small bistable gap. On the other hand, if the bistable gap is increased, the chaotic responses might appear that will result in a significant power reduction at high water flows. Even though at an excessive large bistable gap, i.e. larger than 0.5D, chaotic responses are less likely to occur, the inter-well vibrations are hardly maintained and the improvement in the performance by means of the introduction of the bistable spring will not be acquired. In addition, the paper also reported that a high damping value imposed to the system can eliminate the chaotic vibrations. Unfortunately, an excessively high damping value can diminish the vibrating amplitude and, thus, lower energy to be harvested.

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In this paper, we focus on the case where the bistable system is designed with a significant bistable gap to gain a major effect in improving the power at low velocity flows. However, this might result in chaotic responses at high velocity flows. Therefore, to mitigate the effect, an appropriate chaos controller is designed and analyzed. In this way, the performance of the bistable VIV system at high velocity flows will be compromised and the merit of a bistable system is conserved. Please note that this paper mainly presents the methodology to improve the performance in terms of the working velocity. The parameters in this paper are taken arbitrarily, yet based on realistic reasons, and the presented methodology can be applied for any velocity profiles. Several approaches of chaos control have been proposed, e.g. the OGY method [14–16], the time-delay feedback method [17,18], the fuzzy control method [19], the nonlinear control method [20,21], the non-feedback method [22,23], the adaptive control method [24,25], the proportional feedback method [26,27] and the neural networks method [28,29]. From the energy harvesting point of view, it is desirable to keep the control effort as low as possible since the net power that can be harvested from the system will be reduced by the power required to execute the control effort. Therefore, the OGY controller is preferred as a suitable solution since only a small perturbation on a structural parameter of the system is required to perform the control action [30]. In chaotic dynamical systems, a chaotic attractor is extremely sensitive to initial conditions and unstable periodic orbits are densely embedded within the attractor. Therefore, an appropriate small perturbation to an accessible system parameter will be able to alter the system states. If the perturbation is prescribed properly, stabilization of the system to follow a desired periodic orbit is also feasible [14]. Furthermore, in a practical circumstance where prior analytical knowledge on the system is unavailable and the measurable variables are limited, the OGY method can also be applied by utilizing the time-delay coordinates [31,32]. In this paper, design and implementation of an OGY controller for a bistable VIV energy harvesting system to stabilize its chaotic vibrations by maintaining the responses in periodic inter-well vibrations are presented. In turn, the controller expectedly will improve the utilizable power. The bistable VIV system is modelled as a 4-dimensional autonomous continuous-time dynamical system through a wake oscillator model. Since the OGY controller was initially introduced for a discrete-time system, therefore, the data from the bistable VIV system will be discretized at a certain period according to the Poincaré sectioning. Based on this map, critical information of the system required for the controller such as the fixed points, the Jacobian and sensitivity matrices can be estimated. For a periodically driven dynamical system, the Poincaré map is commonly constructed naturally based on the period of the driving term. However, the bistable VIV system is an autonomous system, where an explicit-time-dependent driving term does not exist and no implementation of such controller on a high-dimensional autonomous system can be found in the literature. Therefore, to overcome this difficulty, an (N–1)-dimensional hyperplane transversal to the chaotic trajectory in RN, where N denotes the dimensions of the system, will be utilized to estimate a period to discretize the continuous system. Subsequently, the fixed points that correspond to the desired periodic orbits can be estimated by the recurrence method. Afterwards, the least square regression method will be utilized to estimate the Jacobian and sensitivity matrices. Finally, the perturbation in the regulated parameter is prescribed based on the control gain matrix achieved through the pole placement technique. A testing platform that comprises a mechanical structure and a computer-feedback system is initially developed to confirm the occurrence of chaotic responses on the system [10]. The structure interacts with exerting forces (e.g. fluid force generated from the fluid vortices) that are measured by a force sensor. The measured forces are fedback to a computer to calculate the instantaneous displacement of the structure based on the simulation model. Subsequently, the calculated displacement is prescribed to manipulate the structure motion through an appropriate position controller. In the later part, to validate the proposed chaos control strategy in a real-time environment, the same testing platform is used. However, for the latter purpose, in order to isolate the exerting force from any potential disturbances, we use a simulated exerting force from the wake oscillator model. In the following, Section 2 discusses some experimental observations from a bistable VIV system in real water tunnel that will support the designing procedure for the controller. Section 3 presents the wake oscillator model to model a vortex induced vibration system. A bifurcation analysis to select critical parameters applied to the model will also be discussed in this section. Subsequently, Section 4 elaborates the design and implementation procedure of the OGY chaos controller and discusses how the controller can be adapted to the bistable VIV system. The simulation results achieved from the proposed controller are presented in Section 5. An experimental validation to confirm the applicability of the controller in realtime situations is discussed in Section 6 and some appropriate conclusions are drawn in Section 7.

2. Observations from experiment A typical VIV system for energy harvesting can be modelled as shown in Fig. 1. The essential component of the system is a circular cylinder with a diameter of D and length of L, which is submerged into a water flow with its length oriented along the z-axis. The cylinder is elastically supported by linear springs with the equivalent stiffness of k, where its motions are constrained only in one degree-of-freedom along the y-axis. In this arrangement, when the water flow crosses over the cylinder in the x-axis at the flow velocity of U, the structure will oscillate only in the y-axis. The displacement of the cylinder in this direction is referred to as y. The effective mass of the oscillating structure, mosc, comprises of the cylinder mass and one third

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Fig. 1. Model of a VIV energy harvesting system (reproduced from [10]).

of the mass from the springs. The total damping factor, c, comprises the damping from the PTO component, e.g. gear box, generator and electrical load, and structural damping from other mechanical components. In this study, a bistable spring substitutes the linear spring on the VIV system. The left panel of Fig. 2 illustrates the schematic of a bistable spring mechanism. Compared to a linear spring that only comprises a single stiffness value of k1, that results in a stable fixed point at the origin, an additional set of springs, k2, provides a nonlinear spring component in the y-axis and allows for two stable fixed points, while the initial fixed point at the origin turns into an unstable fixed point. The equivalent restoring force function of this mechanism in the y-axis, Fk, can be derived by breaking down the restoring forces from the two horizontal springs k2 into this axis (see [33] for detailed discussion). In this study, the bistable restoring force function will be utilized in its idealized piecewise form (Eq. (1)) that retains the two most critical parameters of a bistable stiffness factor, i.e. equivalent stiffness, k, and bistable gap, b, as shown in the right panel of Fig. 2.

F k ¼ k½y  signðyÞb

ð1Þ

As discussed in Section 1, a structural parameter that can be externally manipulated is the most critical part in the implementation of the OGY controller, which includes effective mass of the oscillating structure, spring parameters and damping factor. However, from practical point of view, since the electrical load contributes to the total damping factor of the system [34], therefore, the damping factor is selected as the perturbed parameter on the controller. Manipulating the electrical load also brings additional advantage as it will involve minimum physical interaction on the system. An experiment of a VIV system in a real water tunnel facilitated by a computer-based forced-feedback testing platform (see [10,11] for detailed discussion on the setup) is carried out to confirm the effect of the damping factor on the bifurcation in the practical circumstance. The schematic diagram of the setup is illustrated in Fig. 3. In this setup, a circular cylinder is immersed into a water tunnel that can deliver a steady water flow ranging from 0.02–0.7 m/s. The cylinder is connected to a belt-drive actuator through a tension/compression force sensor. The belt-drive actuator drives the system to translate along the cross-flow direction. In this arrangement, when the water flow crosses over the cylinder, the instantaneous induced fluid force in the cross-flow direction will be measured by the force sensor. The measured induced fluid force, Fmeasured, is acquired to a computer-based controller where the structural parameters of the VIV system, i.e. effective mass, total damping and supporting stiffness, are virtually prescribed. The following governing equation is numerically evaluated to find the instantaneous values of the cross-flow displacement, y.

€ þ cv irtual y_ þ kv irtual y ¼ F measured mv irtual y

Fig. 2. Bistable spring mechanism and the idealized bistable spring force function (reproduced from [10]).

ð2Þ

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1101

Fig. 3. Testing platform and the components: circular cylinder, force sensor, belt-drive actuator and water tunnel (reproduced from [10]).

The analyzed instantaneous displacement values are utilized to control the position of the cylinder through the belt-drive actuator. In short, the cylinder will be displaced by the induced fluid force governed by the model. When the cylinder is moving, the subsequent instantaneous induced fluid force will be measured by the force sensor and fed-back to the controller to evaluate the corresponding displacement state, repetitively. This process is iterated at a high frequency and results in an experimental-numerical VIV phenomenon. The bistable restoring force in Eq. (1) can be easily substituted for the restoring force term in Eq. (2) for the embedment of bistable stiffness factor into the system. Fig. 4 shows the periodic and chaotic time series displacement responses of the system with different values of the total damping and water flow velocity. Since the responses are noise-contaminated, the distinction of periodic and chaotic responses can be carried out by using a surrogate data testing procedure [10]. Fig. 4(a) shows that the responses are periodic under the two different water flows. However, when damping of the system varies, the system might bifurcate from periodic vibrations to chaotic vibrations (see Fig. 4(b)). This observation shows that the damping factor has an evident contribution to the rise of the chaotic responses in the system. 3. Wake oscillator model for modelling vortex induced vibration system and system bifurcation analysis To study the chaotic responses and to design an appropriate controller on the bistable supported VIV system, a wake oscillator model is utilized. This model, which was proposed in [35] and modified by Farshidianfar and Dolatabadi [36] with the additional nonlinear terms, couples the dynamics equation of the VIV structure in 2D form (Eq. (3)) and the wake oscillator equation (Eq. (4)).



   1 2pStU 1 € þ rs þ c ms þ pC M qD2 y qD2 y_ þ hy ¼ qU 2 DC L0 q 4 D 4

ð3Þ



2 2pStU 2pStU A €þe € q ð1  bq2 þ kq4 Þq_ þ q¼ y D D D

ð4Þ

where ms: effective mass of oscillating structure per unit length, CM: added-mass coefficient, q: density of water, rs: total damping coefficient per unit length, c: stall parameter (c = CD/(4pSt) where CD denotes drag coefficient), St: Strouhal number, h: supporting stiffness per unit length, CL0: reference lift coefficient, q: wake variable, e: coefficient of nonlinearity, b: damping coefficient for second-order term over e, k: damping coefficient for fourth-order term over e, A: force coefficient. In this 2D model, ms, rs and h correspond to mosc, c and k, respectively. To simulate the bistable supported VIV system, the bistable spring force function in Eq. (1) substitutes the linear restoring force term in Eq. (3), which results in:


  
 1 2pStU 1 € þ rs þ c ms þ pC M qD2 y qD2 y_ þ h½y  signðyÞgD ¼ qU 2 DC L0 q 4 D 4

ð5Þ

where g denotes the normalized bistable gap to the cylinder diameter, D. To represent the equations of motion of the system in state-space form, the following state vector is defined:

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Fig. 4. Time series displacement responses of the system with different values of the total damping and water flow velocity.

_ T x ¼ ½x1 x2 x3 x4 T ¼ ½y y_ q q

ð6Þ

The governing equations (Eqs. (4) and (5)) are now can be rewritten in state-space form as:

x_ 1 ¼ x2 !
   1 2pStU 4 2 2 _x2 ¼ qU DC L0 x3  ½rs þ c qD x2  h½x1  signðx1 ÞgD 4 D 4ms þ pC M qD2 x_ 3 ¼ x4 !  
   A 1 2pStU 4 2 2 qU DC L0 x3  rs þ c qD x2  h½x1  signðx1 ÞgD x_ 4 ¼ D 4 D 4ms þ pC M qD2

2 2pStU 2pStU ð1  bx23 þ kx43 Þx4  x3 e D D

ð7Þ

Bifurcation maps are generated by simulating the model at varying system parameters of interest to study the nature of the chaotic responses in the system. The upper panel of Fig. 5 shows a ‘bifurcation-like’ diagram in terms of the utilizable power, P, generated from the vibrations of a bistable system and a linear system when the water flow velocity, U, is varied in the operating range. To have a direct comparison between the two systems, the same equivalent supporting stiffness, h, is considered, while to simplify the analysis of the utilizable power, the energy dissipated by the damping component in the system (see Eq. (8)) is used since the damping from the electrical load is included in the total damping term, rs.

P¼

rs t2  t1

Z

t2

_ dt ðyÞ 2

ð8Þ

t1

It can be observed that at low water flows, the bistable system can significantly improve the utilizable power. This merit is advantageous when the water flow that generates the vibrations is drifting towards low water flows. Unfortunately, when the water flow velocity is increasing, the utilizable power from the bistable system noticeably drops. This reduction in power is resulted from the chaotic vibrations as indicated by the bifurcation map shown on the lower panel of Fig. 5. This motivates the implementation of the OGY controller to mitigate the chaotic problem and to improve the performance of the bistable system when the water flow velocity varies. Water flow of 0.16 m/s, which results in chaotic responses, is chosen for the controller study and validation in this paper. As a complement to the analysis of the abovementioned chaotic bifurcation, Fig. 6 shows the bifurcation map of the system for varying total damping, rs, simulated at the water flow of interest (=0.16 m/s). It can be observed that when the

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Fig. 5. Utilizable power, P, of the linear system and the bistable system (upper) and the bifurcation map on vibrating amplitude of the bistable system (lower) as a function of water flow velocity, U.

Fig. 6. Bifurcation map on vibrating amplitude of the bistable system as a function of the total damping factor.

damping is low, the system exhibits well-behave responses in the inter-well vibrating mode. However, when the VIV structure is connected to the PTO component, the damping increases and the responses become chaotic. Increasing the damping further, as implied by the map, the chaotic responses are suppressed; however, the inter-well vibrations are not maintained either. This means the system is not operating effectively at high damping values. For validating the proposed controller, a moderate damping value is chosen as indicated in Fig. 6. At this damping value, the system exhibits chaotic responses and the OGY controller will be designed and implemented to mitigate the problem. The structural parameters and other relevant parameters of the wake oscillator model are listed in Table 1. 4. OGY controller for an autonomous continuous-time dynamical system The OGY controller is developed based on the two fundamental characteristics of a chaotic attractor, i.e. (i) chaotic response is sensitively dependent on the initial conditions and (ii) chaotic attractor contains a large number of unstable

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B.H. Huynh et al. / Mechanical Systems and Signal Processing 98 (2018) 1097–1115 Table 1 Structural parameters and other parameters of the wake oscillator model. VIV structure

Wake oscillator model

Parameters

Values

Parameters

Values

D ms rs h g

0.05 m 19.8313 kg/m 17.8067 Ns/m2 421.6078 N/m2 0.075

CM St CL0

1 0.2 0.3 0.3 0.0625 0.0005 12 2.55 0.16 m/s 1000 kg/m3

e b k A CD U

q

periodic orbits. Therefore, by applying a proper perturbation on the system, it is feasible to steer the attractor towards a desired orbit among those unstable periodic orbits. Descriptions and applications of the OGY chaos controller are comprehensively discussed in [37,38]. The OGY controller was first introduced to a discrete-time N-dimensional dynamical system:

xnþ1 ¼ Fðxn ; pn Þ

ð9Þ

where xn 2 RN, F denotes a smooth vector function and pn is an accessible parameter of the system which can be externally regulated. Referring to the bistable VIV system in Eq. (7), the system has N = 4. To apply the controller for a continuous-time dynamical system, this system will be discretized following Eq. (9) by the utilization of a Poincaré map that is constructed at a certain period, which basically is the intersection of orbits in the state space to a lower dimensional (hyper-)plane. , the system F(xn, p ) manifests itself as a chaotic Assuming that at a nominal value of the accessible parameter, pn ¼ p attractor, the objective of the controller is to perturb the parameter pn in such the dynamics of the system converge to a desired periodic orbit embedded in the attractor for almost all initial conditions in the basin of the attractor. To avoid large control efforts, the perturbation in the regulated parameter is constrained in a narrow range, d, as:

j < d jpn  p

ð10Þ

Considering a small neighborhood of the desired period orbit, the dynamics of the system can be linearized and if the controllability is guaranteed, the system can be stabilized. The nature of a chaotic/strange attractor dictates that the state trajectory always enters the neighborhood. Therefore, it is rational to assume that the trajectory can be steered and stabilized to a selected periodic orbit by a feedback control. When a continuous-time dynamical system is discretized at a certain period, a period one orbit (a period that corresponds to a single point in the Poincaré section that repeats itself in every cycle) can be assumed to be a fixed point of the discrete Þ, that satisfies the following condition: map F, x ðp

Þ ¼ xn ðp Þ ¼ x ðp Þ xnþ1 ðp

ð11Þ

Þ, that satisfies: Consequently, a period M orbit corresponds to a fixed point, x ðp

Þ ¼ xn ðp Þ ¼ x ðp Þ xnþM ðp

ð12Þ

From the perspective of energy harvesting, a bistable VIV system takes advantage most when it operates at the inter-well vibrating mode, where the trajectory corresponds mostly to the period one orbit. Therefore, the desired periodic orbit in this case is the period one orbit of the discrete map as reflected in Eq. (11). If the dynamics of the system is within the defined neighborhood of the fixed point with the accessible parameter closed , the map can be linearized as: to the nominal value, p

Þ ¼ A½xn  x ðp Þ þ B½pn  p  xnþ1  x ðp

ð13Þ

where A denotes the N  N Jacobian matrix and B is an N  1 matrix that represents the effect of the regulated parameter pn  as: or the sensitivity matrix. The matrices A and B are evaluated at xn ¼ x and pn ¼ p

@ Þ Fðx ; p @xn @ Þ Fðx ; p B¼ @pn

A¼

ð14Þ

Þ and the state xn as: Assuming a linear correlation between the perturbation in the regulated parameter ðpn  p

 ¼ K T ½xn  x ðp Þ pn  p

ð15Þ

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Therefore, substituting Eqs. (15)–(13), the following relationship is obtained:

Þ ¼ ðA  BK T Þ½xn  x ðp Þ xnþ1  x ðp

ð16Þ

If the eigenvalues of matrix (A – BKT) are smaller than unity, saying that the matrix is asymptotically stable, the fixed point is confirmed to be stable. The matrix KT can be solved by using the pole placement technique through the Ackermann’s method [39] with the prerequisite of a full rank (=N) controllability matrix, C, which is an N  N matrix:

. . . . C ¼ bB..AB..A2 B..    ..AN1 Bc

ð17Þ

Þ is an unstable fixed point, the matrix A is characterized by Nu unstable eigenvalues and Ns stable eigenvalues, Since x ðp where Nu + Ns = N. To achieve the super-stability in the pole placement technique, we keep the Ns poles of the matrix A, while the remaining Nu poles are set to zero. Referring to Eqs. (10) and (15), it can be immediately seen that:

Þj < d jK T ½xn  x ðp

ð18Þ T

This means, the control law is enabled only when the trajectory of the system is within the neighborhood of 2d=jK j in RN . space, otherwise, the regulated parameter remains as its nominal value, p 4.1. Construction of a Poincaré map for an autonomous continuous-time dynamical system In order to implement the OGY controller for a continuous-time dynamical system, the first and foremost step is discretizing the system through the Poincaré map. Based on the constructed Poincaré map, relevant information of the system can be acquired that includes fixed points and the state space matrices A and B. For a periodically driven dynamical system, there exists a natural Poincaré map that corresponds to the period of the driving term. Therefore, the system can be simply discretized at the corresponding period (see e.g. [30,40,41]). However, the bistable VIV system (Eq. (7)) is an autonomous system where an explicit-time-dependent driving term does not exist. Therefore, the period to discretize the system must be estimated. In this study, the period to discretize the system, T, is estimated by using the method discussed in [42], which is based on the intersection of the phase plane trajectory at an (N–1)-dimensional hyperplane in RN space. The system model in Eq. (7) is simulated at a very fine fixed time step for sufficiently long duration to gather a comprehensive map. Having a representative phase plane trajectory, the state vector x is sampled every time the trajectory intersects a lower dimensional hyperplane (N– 1). Obviously, in a case of a well-behaved system, the trajectory will intersect the intersecting hyperplane at the same position every time it pierces the plane. However, for a case of chaotic attractor, the piercing points at the hyperplane will be scattered within a closed area. Therefore, if there exist two consecutive state vectors x(t1) and x(t2) on the intersecting hyperplane, where their Euclidian distance is smaller than a certain threshold, then the period is defined as T = t2 – t1. If there are many pairs of consecutive state vectors that satisfy this condition, the estimation of the period is averaged. To maximize the harvested energy, the inter-well period one orbit is desired. Therefore, in order to ensure that the orbit caters for the inter-well vibration, intuitively, the discretizing period is taken when the displacement crosses the unstable fixed point of the double-well bistable spring. 4.2. Estimation of fixed points After the period T to discretize to system is achieved, the next step in the designing procedure is to estimate the fixed Þ that satisfies xnþ1 ðp  Þ ¼ xn ðp Þ ¼ x ðp Þ must points of the discrete map. Since a period one orbit is desired, the fixed point x ðp be found. Such a fixed point can be estimated by the recurrence method proposed in [43]. Since a period one orbit is assumed to be embedded in the chaotic attractor of the continuous-time dynamical system, a Þ that is nearby to this orbit will remain in the neighborhood of this orbit in, at least, one period of the orbit. state vector xn ðp Þ and Therefore, the trajectory of such state vector can be regarded as the unstable orbit. In other words, the state vectors xn ðp Þ that are in the vicinity of each other can be considered as an estimation of the fixed point x ðp Þ. xnþ1 ðp In order to estimate this fixed point, first, a large amount of state vectors or data points is generated by integrating the Þ that satisfy Eq. (19) are searched. system (Eq. (7)) at the estimated period T for certain duration. Next, the state vectors xn ðp

Þ  xn ðp Þk < j kxnþ1 ðp

ð19Þ

where j defines a vicinity or threshold in which the state vector is assumed to remain the same after one period. The state vectors that satisfy Eq. (19) are called recurrent state vectors. The longest series of state vectors, Þ; xnþ1 ðp Þ; xnþ2 ðp Þ; xnþ3 ðp Þ; . . ., where all the elements are the recurrent state vectors, has the highest possibility to rep½xn ðp Þ, the state vectors in this series and all other neighresent the desired fixed point. Therefore, to estimate the fixed point x ðp boring recurrent state vectors within the vicinities of j are averaged.

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4.3. Estimation of matrices A and B Þ estimated by the recurrence method, the matrices A and B in Eq. (13) can be estimated by Based on the fixed point x ðp the least square regression method discussed in [44]. To estimate the matrix A, the set of state vectors that was used to estiÞ; xnþ1 ðp Þ that satisfy the following equation are mate the fixed point can be reused. First, the pairs of state vectors ½xn ðp searched.

Þ  x ðp Þk < v kxn ðp Þ  x ðp Þk < v kxnþ1 ðp

ð20Þ

where v defines a vicinity or threshold in which the linearization in Eq. (13) is valid. The linear fit for the function Þ  x ðp Þ ¼ f ½xn ðp Þ  x ðp Þ can be expressed as: xnþ1 ðp

Þ  x;1 ðp Þ ¼ a11 ½xn;1 ðp Þ  x;1 ðp Þ þ a12 ½xn;2 ðp Þ  x;2 ðp Þ þ ::: þ a1N ½xn;N ðp Þ  x;N ðp Þ þ c1 xnþ1;1 ðp Þ  x;2 ðp Þ ¼ a21 ½xn;1 ðp Þ  x;1 ðp Þ þ a22 ½xn;2 ðp Þ  x;2 ðp Þ þ ::: þ a2N ½xn;N ðp Þ  x;N ðp Þ þ c2 xnþ1;2 ðp .. .

ð21Þ

Þ  x;N ðp Þ ¼ aN1 ½xn;1 ðp Þ  x;1 ðp Þ þ aN2 ½xn;2 ðp Þ  x;2 ðp Þ þ ::: þ aNN ½xn;N ðp Þ  x;N ðp Þ þ cN xnþ1;N ðp The matrix A is therefore defined as:

2

a11

a12

aN1

a22 .. . aN2

6a 6 21 A¼6 6 .. 4 .

...

a1N

3

. . . a2N 7 7 .. .. 7 7 . . 5    aNN

ð22Þ

Þ; xnþ1 ðp Þ that satisfy Eq. (20), the matrix A can be calBy collecting a sufficient amount of pairs of the state vectors ½xn ðp culated by the least square regression method. It should be noted that the matrix [c1 c2 . . . cN]T in Eq. (21) denotes an offset that can be neglected if the estimation of the fixed point is sufficiently accurate. To estimate the matrix B that represents the effect of the regulated parameter pn, another set of state vectors must be generated. The system shown in Eq. (7) is integrated at the period T for certain duration with the maximum perturbation, d, activated at every alternating steps, i.e. the maximum perturbation is enabled at the time point n and disabled at the time Þ; xnþ1 ðp  þ dÞ. In order to ensure the validity of the linearization in Eq. (13), point (n + 1) to form a pair of state vectors ½xn ðp the pairs of the state vectors that satisfy the following equation are explored:

Þ  x ðp Þk < v kxn ðp  Þk < v kxnþ1 ðp þ dÞ  x ðp

ð23Þ

Then, the matrix B can be calculated by the following relationship:

2  þ dÞ  x;1 ðp Þ 3 Þ  x;1 ðp Þ 3 xn;1 ðp xnþ1;1 ðp 6 xnþ1;2 ðp 6 xn;2 ðp  þ dÞ  x;2 ðp Þ 7 Þ  x;2 ðp Þ 7 7 7 6 6 7 ¼ A6 7 þ Bd 6 . .. 7 7 6 6 . 5 5 4 4 . . 2

 þ dÞ  x;N ðp Þ Þ  x;N ðp Þ xnþ1;N ðp xn;N ðp 02 2  þ dÞ  x;1 ðp Þ 3 Þ  x;1 ðp Þ 31 xn;1 ðp xnþ1;1 ðp C B6 xnþ1;2 ðp 6 xn;2 ðp  þ dÞ  x;2 ðp Þ 7 Þ  x;2 ðp Þ 7 B6 7 7C 1 6 7  A6 7C 6 )B¼B .. B6 7 7C d 6 ... @4 5 5A 4 .  þ dÞ  x;N ðp Þ xnþ1;N ðp

ð24Þ

Þ  x;N ðp Þ xn;N ðp

Many pairs of the state vectors that satisfy Eq. (23) provide the averaged calculation of the matrix B. Þ, the matrices A and B are estimated, the design and implementation of the OGY chaos controller After the fixed point x ðp can be executed following the procedure discussed in Section 4.1. Þ, matrices A To simplify the selections of the thresholds j and v, for estimating the discretizing period T, fixed point x ðp and B, the Euclidian distance of two state vectors, e.g. xn and xn+1, is normalized to the distance range of the corresponding state element:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N  2 uX xnþ1;i  xn;i kxnþ1  xn k ¼ t maxðxi Þ  minðxi Þ i¼1

ð25Þ

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Þ and the matrices A and B are dependent on the It should be noted that the accuracies in the period T, the fixed point x ðp selections of the thresholds j and v and the number of state vectors in the data sets. Therefore, a trial and error attempt may be required regarding the selections of the thresholds j and v and the number of state vectors in the data sets in order to ensure the performance of the OGY controller. 5. Simulation results and discussions This section discusses the OGY controller designed for the system with parameters shown in Table 1. The development of the controller follows the procedure discussed in Section 4 that is initiated by the discretization of the model continued to the determination of the discrete state-space equation. Estimation of the discretizing period T The model is simulated at a fixed time step, Dt = 0.5 ms, collected for long duration (200,000 s) in order to have a representative chaotic time series. This time series is generated to estimate the discretizing period by intersecting the chaotic attractor to a subspace hyperplane section (Poincaré). Fig. 7 shows the corresponding time series of the cylinder displacement. From the intersecting section, the discretizing period is estimated at T = 1.6575 s. Estimation of the fixed point Generating state vectors at the Poincaré section that results in about 2,000,000 data points is carried out to estimate the Þ, through the recurrence method. Fig. 8 shows the location of the estimated fixed point, which is obtained fixed point, x ðp from the longest identified series of recurrent points with a neighboring threshold j = 0.0003, on the Poincaré section. The left panel of Fig. 8 illustrates the magnified Poincaré section that contains the estimated fixed point and its neighbors. The estimated fixed point, which later on will be used as the controller target, is identified at x = xtarget = [0.0040–0.0340–13. 1434 16.0563]T. Estimation of Jacobian matrix A Following Eq. (20), pairs of state vectors from the same data set are explored, which later on after the implementation of the least square regression method (Eq. (21)), the Jacobian matrix A is derived:

2

0:4284

0:0970

6 2:8297 0:7412 6 A¼6 4 27:8646 19:3578 501:1878 102:9648

4:1671  104 7:6704  104 0:4271 0:7780

8:1153  105

3

1:4834  104 7 7 7 5 0:0817 0:7398

Derivation of matrix B Selecting the damping parameter, rs, as the regulated parameter, pn, another data set (about 2,000,000 data points) is generated by introducing maximum perturbation d = drs max at every alternating steps. Attaining the pairs of state vectors that satisfy Eq. (23) and matrix B can be evaluated respecting Eq. (24): B = [3.7430  106 0.0013 0.0528 0.0919]T

Fig. 7. Time series displacement of the cylinder (evaluated at the fixed fine time step, Dt = 0.5 ms).

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Fig. 8. Location of the estimated fixed point on the Poincaré section.

Controllability and control matrix The control matrix, KT, can be derived by respecting the controllability condition (Eq. (17)) and implementing the Ackermann’s method. The matrix KT is, therefore, obtained as: KT = [4.3016  103 1.0298  103 –2.1660–0.8316] Controller implementation The control action is implemented through perturbation in the damping parameter, rs, evaluated in Eq. (15). Fig. 9 shows the system response before and after the controller activation. The maximum perturbation, d, is set at drs max = ±5%. The displacement of the system, x1, at the period one orbit is shown on the upper panel and the perturbation in the damping is shown on the lower panel of Fig. 9. The initial condition is selected arbitrarily within the basin of the chaotic attractor at x0 = [0 0 0 50]T. After wandering within the attractor in about 785 s, the state vector approaches the vicinity of the estimated fixed point and the condition in Eq. (18) is achieved. The controller is activated to allow the perturbation in the damping. After the transient response decays in about 140 s, the system reaches the steady state condition and the period one inter-well trajectory is achieved as illustrated in the time series plot on Fig. 10 with the corresponding phase plot on Fig. 11. The achieved fixed point at the steady state is xachieved = [0.0040–0.0341–13.1603 15.9788]T with a steady error

Fig. 9. Controlling result of the system operating at U = 0.16 m/s: values of x1 at every discrete steps (upper); perturbations in the damping at every discrete steps (lower).

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Fig. 10. Time series displacement when the OGY chaos controller is activated.

Fig. 11. Phase portraits of the uncontrolled system and controlled system.

e = [0.000 0.0001 0.0169 0.0775]T. When the steady state of the system is reached, the perturbation becomes steady at drs = 0.3926%. This result is practically beneficial since for a constant perturbation the net harvested power will not be affected by the control effort.

5.1. Transient response improvement As shown in Fig. 10, the system requires about 785 s to reach the fixed point from the initial condition. In order to improve the speed, the maximum perturbation in the regulated parameter can be set larger to cover a wider search area of the fixed point for the system to enter. In this section, a new larger maximum perturbation, drs max = ±10%, is prescribed to the system to study the effect on the transient response. Implementing the same procedure carried out in the previous study, the system response can be seen as in Fig. 12. As shown in the figure, the new controller also demonstrates satisfactory steady-state performance, where the response is clearly stabilized once the system reaches the steady-state condition. However, compared to the result of the controlled system with drs max = ±5%, the new controller results in a faster response, where it only takes about 330 s before the system is becoming stable (see Fig. 9). This response is much faster than that of the system controlled with drs max = ±5%, where it takes about 785 s to stabilize the system. In addition, a larger perturbation in the regulated parameter is required by the system to approach the desired fixed point when the system is controlled with a larger maximum perturbation range, d, as reflected in the lower panels of Figs. 9 and 12. Furthermore, selecting a larger maximum perturbation range, d, might increase a chance of having a false control effort. It can be observed from the lower panel of Fig. 12 that at t  240 s, the system suggests a false control effort when the state vector is x = [0.0040 0.0345 13.8690–10.0855]T, which is relatively far from the targeting fixed point. In this case, the system will not result in a stable state, while such a false control effort does not take place when the system is controlled with drs max = ±5%. Therefore, when the OGY controller is implemented, the maximum perturbation in the regulated parameter, d, must be carefully chosen. The system requires a longer time to approach the desired fixed point when a small value of d is utilized. On the other hand, high value of d might result in false control efforts and the large perturbation in the regulated parameter may lead to a significant change in the structural properties and instability of the system.

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Fig. 12. Controlling result with drs

max

= ±10%: values of x1 at every discrete steps (upper); perturabations in the damping at every discrete steps (lower).

5.2. Verification at different water flow As another validation, this subsection presents the implementation of the OGY controller at a different water flow velocity, U = 0.165 m/s. This analysis is performed to validate further the performance of the controlled system, since in the real operating condition the water flow may fluctuate significantly. In this numerical experiment, the discretizing period T, the fixed point x , the matrices A, B and KT are evaluated as: T = 1.6155 s x = xtarget = [0.0047–0.0366–13.4119 15.1856]T

2

0:4195 6 0:6016 6 A¼6 4 150:4246 3

1:0234  10 5

0:0863 0:1185

6:7565  104 0:0067

27:0696

0:8205

215:6480

3:8386

3 5:6066  105 4 7 4:1032  10 7 7 5 0:0377 0:2772

T

B = [7.7214  10 0.0011 0.0443–0.0082] KT = [5.9726  103 1.1836  103 –16.0066–1.8959] The maximum perturbation in the regulated parameter is also re-selected to accommodate to the variation in nature of the chaotic attractor when the water flow velocity is varied. A trial and error iteration results in the maximum allowable perturbation of the damping parameter at drs max = ±20%. The response of the system under the corresponding controller implementation is shown in Fig. 13. When the steady state is reached, the perturbation in the damping is drs = 13.6206%. The period one inter-well orbit is achieved with the fixed point xachieved = [0.0049–0.0420–13.7370 14.0827]T that results in a steady state error e = [0.0002 0.0054 0.3251 1.1029]T. It should be noted that this error is more significant than one from the previous case of U = 0.16 m/s. This difference in the steady state errors can be explained by the difference in the accuracies in the estimations of the fixed points in two cases. In the case of U = 0.16 m/s, the period one fixed point was defined with the neighboring threshold j = 0.0003, while due to the chaotic nature in this case, the threshold is increased to j = 0.02 to find the fixed point. High neighboring threshold, j, will result in a lower estimation accuracy of the fixed point, which in turn will cause lower accuracies of matrices A and B. As a consequence, a relatively high steady state error is obtained. This concludes that the controller performance is dependent on the accuracy of the fixed point estimation. 5.3. Verification on different system parameters To validate further the applicability of the controller, the OGY is implemented on the same system with different set of structural parameters, i.e. bistable gap g = 0.1 and damping value rs = 16.8174 Ns/m2, where at water flow of 0.13 m/s, this system results in chaotic responses. Implementing the same procedure presented in Section 4, the discretizing period T, the fixed point x , the matrices A, B and KT are estimated as follows:

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1111

Fig. 13. Controlling result of the system operating at U = 0.165 m/s: values of x1 at every discrete steps (upper); perturabations in the damping at every discrete steps (lower).

Fig. 14. Controlling result of the system with g = 0.1 and rs = 16.8174 Ns/m2 operating at U = 0.13 m/s: values of x1 at every discrete steps (upper); perturabations in the damping at every discrete steps (lower).

T = 2.0202 s x = xtarget = [0.0028–0.0290–13.9599 10.7525]T

2 6 6 A¼6 4

1:2852

0:2129

5:5180  104

1:8886

0:0048

0:0050

75:6520

11:0386

0:7307

1:2560  103

284:0342

1:9948

4

T

B = [4.0032  10 0.0016 0.1296–0.0632] KT = [2.1858  103 277.4965 2.5571–0.0960]

1:1132  104

3

2:1266  105 7 7 7 5 0:0480 0:3513

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Table 2 Comparison in utilizable power, P (mW/m), from the uncontrolled system and controlled system at different structural parameters and water flow velocity, U. Parameters Utilizable power

Power of uncontrolled system (chaotic), PU Power of controlled system (periodic), PC Improvement, (PC – PU)  100/PU

U = 0.16 m/s rs = 17.8067 Ns/m2 g = 0.075

U = 0.165 m/s rs = 17.8067 Ns/m2 g = 0.075

U = 0.13 m/s rs = 16.8174 Ns/m2 g = 0.1

10.7267 12.9955 21.15%

9.8233 16.9991 73.05%

6.9653 8.6689 24.46%

Fig. 15. Real-time stategy of the OGY chaos controller implemented to the testing platform.

The system responses before and after the controller activation are shown in Fig. 14. When the allowable maximum perturbation in the damping is set to drs max = ±15%, after the steady state is reached, the system is stabilized at xachieved = [0.0033–0.0290–13.9205 10.4002]T with the constant perturbation drs = 7.171% and a small steady state error e = [0.0005 0.0000 0.0394 0.3523]T. In particular, it only takes about 280 s to enable the control law and additional 195 s for the system to completely reach its steady state. This experiment concludes the applicability of the controller for different systems with different structural parameters, e.g. bistable gap and damping factor. As a summary, Table 2 lists the effect of the controller to the system in term of the utilizable power that is calculated by Eq. (8). It can be observed that in all cases of different values of structural parameters and water flow velocity, the harvested power is significantly improved when the OGY controller is applied. In particular, for the case of U = 0.165 m/s, the power of the controlled system is improved up to 73.05% compared to that of the uncontrolled system. This significant improvement can be understood since the controller stabilizes the system into its period one inter-well orbit, which corresponds to the highest-energy orbit of a bistable system. The improvement in the system performance with the OGY controller shows a promising opportunity for practical applications of bistable VIV energy harvesting systems. It should be noted that in some cases, e.g. cases of U = 0.16 m/s and U = 0.165 m/s, the utilizable power of the controlled system might still be lower than that of the system with linear spring. However, the negative effect from chaotic responses of

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Fig. 16. Time series displacement of the cylinder (upper panel) and the perturbation in the damping (lower panel) when the OGY controller is implemented in real-time.

the bistable system at high velocity flows is suppressed by the chaos controller. This can be considered as a compromise when a bistable system is applied to improve the performance at low velocity flows.

6. Real-time experimental validation An experimental validation is carried out to confirm the feasibility of the OGY controller in a real-time application. The experiment is conducted on the same the testing platform described in Section 2. However, to isolate the uncertainty in the water flow produced by the water tunnel, the wake oscillator model is still utilized to simulate the fluid force exciting the virtual bistable VIV structure. The (virtual) bistable VIV system in Eq. (7) with the OGY controller is simulated in a real-time computer-based forcedfeedback testing platform. The cylinder displacement, y or x1 in its state-space form, will be prescribed to actuate the cylinder motion in real-time. The controller strategy is presented as shown in Fig. 15. The structural parameters used in this experimental validation are the same as those in the simulation for the case of U = 0.16 m/s. The estimated information Þ, the matrices A and B and KT obtained from the previous analysis are including the discretizing period T, fixed point x ðp also utilized for the OGY controller. The control strategy is programmed in the Simulink environment (Matlab R2012a) with the fourth-order Runge-Kutta solver at the fixed time step of 0.5 ms and the control strategy is implemented through the dSPACE 1103 controller. A PID controller is used to position the cylinder in the testing platform, which is actuated through a belt-drive actuator (Toyo M80M-S-X102-400-L-T40-C4) with the repeatability of ±0.08 mm, driven by a 400 W AC servo motor (Delta ECMA-C10604F8) and equipped with a 8192 pulse/rev encoder to measure the real displacement values as well as to form the feedback loop. Fig. 16 shows the time series displacement of the cylinder (upper panel) and the perturbation in the damping (lower panel) when the OGY controller is implemented in real-time. It can be seen that the controller is able to stabilize the system to the period-one inter-well vibrations. When the maximum allowable perturbation is set at drs max = 20%, it takes about 85 s for the controller to approach the desired fixed point and additional 65 s to settle the transient response before the system is completely stabilized. After the steady state condition is reached, a constant perturbation in the damping parameter is drs = 9.2798% and the system is stabilized at the fixed point xachieved = [0.0042–0.0343–13.2650 15.1661]T with a steady state error at e = [0.0002 0.0003 0.1216 0.8902]. In conclusion, the success in stabilizing the system with the OGY controller confirms its feasibility in real-time applications.

7. Conclusions In general, this paper concludes the applicability of a bistable VIV system for energy harvesting. The bistable stiffness on the VIV system is shown to be able to improve the utilizable power at the low water flows. However, chaotic responses may occur and lead to a severe degradation in the harvested power when the water flow velocity varies. To overcome this problem, an OGY controller was designed and implemented to stabilize the chaotic responses and force the inter-well

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periodic vibrations. When the system is controlled, it offers a notably improved harvested power. This result opens a potential window for practical applications of bistable VIV energy harvesting systems. Although the chaos controller can minimize the negative effect of the chaotic responses, in some cases, the harvested power from the controlled bistable system still slightly lower than the level of power from the linear system. However, it has been noted that the objective of introducing the nonlinear element and the proposed controller is to improve the low velocity side with minimum forfeit on the high velocity side. Moreover, the proposed system is also scalable, which means that the system can be adapted according to the natural water flow characteristics. The success and performance of the OGY controller for the bistable VIV system are essentially dependent on the accuracies in the estimation of the period to discretize the system, the fixed points, the Jacobian and sensitivity matrices. A trial and error process may be required to properly select the threshold values, j and v, and the number of state vectors in the data sets in order to improve the performance of the controlled system. The maximum allowable perturbation in the regulated parameter must be rationally chosen in order to allow the system to approach the desired fixed point in reasonable time and to minimize the potential instability in the operation of the system caused by the large control efforts in the transient duration. The applicability of the OGY controller in a real-time application was further confirmed by an experimental validation conducted on the computer-based testing platform. However, it should be highlighted that an extensive study on the system to be controlled is required in designing the OGY controller, i.e. the real-time state vectors of the system operating in a sufficiently-long duration are recorded to estimate the discretizing periods, fixed points, the matrices A and B. Unfortunately, measuring all the state variables of the system is a tedious process. As an alternative, the time-delay coordinates, which are reconstructed from a single state variable, can also be utilized to design an OGY chaos controller. In the next stage of this research, the OGY controller based on the time-delay coordinates will be investigated. Acknowledgements The authors would like to acknowledge: (a) The Interdisciplinary Graduate School, Nanyang Technological University for the research scholarship award, (b) Energy Research Institute at Nanyang Technological University for the technical support, (c) Singapore Ministry of Education for The Academic Research Fund Tier 1, Project title: Nonlinearly enhanced flow induced vibration structure for energy generator – RG106/14 and (d) Maritime Research Centre, Nanyang Technological University. References [1] M.M. Bernitsas, K. Raghavan, Y. Ben Simon, E.M. Garcia, VIVACE (Vortex Induced Vibration Aquatic Clean Energy): a new concept in generation of clean and renewable energy from fluid flow, J. Offshore Mech. Arct. Eng. 130 (2008) 041101. [2] C.H.K. Williamson, R. Govardhan, Vortex-induced vibrations, Annu. Rev. Fluid Mech. 36 (2004) 413–455. [3] T. Sarpkaya, A critical review of the intrinsic nature of vortex-induced vibrations, J. Fluids Struct. 19 (2004) 389–447. [4] R.D. 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