Krauskopf, B

Sieber, J

We demonstrate a method for tracking oscillations and their stability boundaries (bifurcations) in nonlinear systems. Our method does not require an underlying model of the dynamical system but instead relies on feedback stabilizability. Our method allows one to determine bifurcations of the dynamical system without the need to observe the transient oscillations for a long time to determine their decay or growth. Moreover, in the context of hybrid experiments, which couple experiments and computer simulations bidirectionally and in real-time, our method is able to overcome the presence of coupling delays (or, more generally, unknown actuator dynamics), which is a fundamental problem that is currently limiting the use of hybrid testing. We illustrate the basic ideas with a computer simulation (including coupling delays and noise) of a prototype nonlinear hybrid experiment: a real pendulum coupled at its pivot to a computer simulation of a vertically excited mass-spring-damper system

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                          Sieber, J., & Krauskopf, B. (2008). Tracking of nonlinear oscillations inhybrid experiments.Early version, also known as pre-printLink to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTRACKING OF NONLINEAR OSCILLATIONS IN HYBRIDEXPERIMENTSJan Sieber1 and Bernd Krauskopf21 Centre for Applied Dynamics Research (CADR))School of Engineering, University of Aberdeen, King’s College, Aberdeen AB24 3UE, UnitedKingdome-mail: J.Sieber@abdn.ac.uk2 Department of Engineering MathematicsQueen’s Building, Universtity of BristolUniversity Walk, Bristol BS8 1TR, United Kingdome-mail: B.Krauskopf@bristol.ac.ukKeywords: hybrid testing, bifurcation analysis, coupling delay, numerical continuationABSTRACTWe demonstrate a method for tracking oscillations and their stability boundaries (bifurcations)in nonlinear systems. Our method does not require an underlying model of the dynamicalsystem but instead relies on feedback stabilizability. Our method allows one to determine bifur-cations of the dynamical system without the need to observe the transient oscillations for a longtime to determine their decay or growth. Moreover, in the context of hybrid experiments, whichcouple experiments and computer simulations bidirectionally and in real-time, our method isable to overcome the presence of coupling delays (or, more generally, unknown actuator dy-namics), which is a fundamental problem that is currently limiting the use of hybrid testing. Weillustrate the basic ideas with a computer simulation (including coupling delays and noise) ofa prototype nonlinear hybrid experiment: a real pendulum coupled at its pivot to a computersimulation of a vertically excited mass-spring-damper system.1. INTRODUCTIONA common task in the study of nonlinear dynamical systems is to find and track oscillations andtheir properties (amplitude, spectrum, dynamical stability, etc.). As one varies system parametersthese properties can change, for example, the oscillation may become dynamically unstablein a period doubling, or it can “disappear” in a saddle-node. These special events (and theirparameter values) are called bifurcations. When a mathematical model of the system underconsideration is known, for example, in the form of an ordinary differential equation (ODE), thenthe nonlinear oscillations and their bifurcations can be tracked, regardless of their dynamicalstability, in a suitable parameter as periodic solutions of the ODE. This tracking (one also speaksE134of path-following or numerical continuation) of different types of solutions and their bifurcationshas emerged as a tremendously useful tool for the analysis of nonlinear dynamical systemsbecause it reveals the boundaries of parameter regions with qualitatively different long-timebehaviour. Numerical continuation can be performed with a number of freely available softwarepackages; see, for example, the recent survey [1].In this paper we consider the problem of tracking oscillations and stability boundaries directlyin situations when a full mathematical model of the system under consideration is not available,for example in experiments. More specifically, we are interested in hybrid testing experimentsof mechanical and civil engineering systems [2–5]. A hybrid experiment couples a mechanicalexperiment and a computer simulation bidirectionally and in real-time. One major aim of theseexperiments is finding and tracking stability boundaries.We illustrate the basic ideas of control-based bifurcation tracking with a period doublingbifurcation in a prototype nonlinear hybrid experiment: a real pendulum coupled at its pivot to acomputer simulation of a vertically excited mass-spring-damper (MSD) system as sketched inFigure 1. This study is still based on coupling the computer simulation to a computer simulationof the pendulum because the introduction of feedback control requires a change to the physicalsetup of the experiment. The original combined MSD-pendulum system (an auto-parametricallyy yyθ θacos(Ωt)simulation experimentF FF(a) (b)y¨+β y˙+αy=acos(Ωt)+Fyayameasurement:F, θ , yaactuatorFigure 1: Sketch of decomposition of the overall mechanical system into a computer simulationof a mass-spring-damper system and a real pendulum. Panel (a): original (emulated) system,panel (b): hybrid experiment, the bidirectionally real-time coupled system as studied in [4, 5].excited two-degree of freedom oscillator) shows a rich bifurcation structure, which can beexplored systematically with the numerical methods implemented in AUTO [6] and explained in[7]. This makes the MSD-pendulum system an ideal test candidate, both, for our method and forhybrid testing of systems with nonlinear dynamical behaviour in general. In non-dimensionalizedform the MSD-pendulum system shown in figure 1(a) is governed by the equations(1+ p)y¨+β y˙+αy+ p[θ¨ sinθ + θ˙ 2 cosθ ] = acos(Ωt) (1)θ¨ +ζ θ˙ +(1+ y¨)sinθ = 0 (2)where p=m/M is the mass ratio between the mass M of the mass in the MSD subsystem and theeffective mass m of the pendulum. We use the time unit ω−1 =√l/g where g is the accelerationdue to gravity and l is the effective length of the pendulum.E1342. ESSENTIAL INSTABILITIES IN DELAY-COUPLED MECHANICAL SYSTEMSThe basic work cycle in a hybrid experiment consists of three parts (see figure 1(b)):1. the numerical simulation of the model (in the example, y¨+β y˙+αy= acos(Ωt)+F(t))with the force input F(t);2. feeding the actuator with the simulation output y(t), and3. feeding back the force measurement F(t) to the simulation.All parts of the loop need to be implemented in real time, and need to run simultaneously andin parallel to the experiment. One advantage of the hybrid setup is that one can easily andsystematically vary the system parameters of the numerical subsystem (a, Ω, α , and β in ourcase) and still study the experimental (nonlinear) component in its original size.The real-time requirement for the coupling between simulation and experiment via a transfersystem and force measurements introduces a number of difficulties. For example, there will be amismatch between the prescribed trajectory y(t) obtained from the simulation and the outputya(t) of the transfer system; see figure 1(b). This mismatche(t) = ya(t)− y(t) (3)is called the synchronisation error. It is caused by the dynamics of the actuator, which is ingeneral not able to follow the prescribed input trajectory perfectly and instantly. Since the outputof the simulation y(t) is known, the synchronisation error (3) can be determined in a hybridexperiment by recording the actual motion ya(t) of the transfer system (the mechanical actuator).If the synchronisation error e(t) is small then this is taken as a measure of accuracy of the wholehybrid experiment [3, 8].In many situations the actual trajectory ya(t) of the actuator follows the prescribed trajectoryy(t) almost exactly, but with a fixed small pure time delay τ [4, 8, 9]. Hence, the actuator can bemodelled asya(t) = y(t− τ). (4)Let us consider the system in figure 1 in the configuration where the mass m of the pendulum islarger than the mass M of the mass block in the mass-spring-damper system, that is, p> 1 in(1), (2). This configuration was found to be impossible to run as a hybrid experiment (that is,as a coupled system as shown in figure 1 (b); see [4]). The analysis in [4, 10] shows that thisobstruction is due to the small coupling delay τ . The delay τ changes system (1), (2) toy¨+ py¨(t− τ)+β y˙+αy+ p[θ¨ sinθ + θ˙ 2 cosθ ] = acos(Ωt),θ¨ +ζ θ˙ +(1+ y¨(t− τ))sinθ = 0 (5)where we inserted the dynamics of the actuator (4) for ya. In system (5) we have dropped theargument t from all dependent variables y and θ except for those that feature a delay. Theappearance of a delayed highest derivative makes (5) a neutral delay differential equation [11].How does system (5) behave if the mass ratio p is larger than one and the delay is small?The interaction between a nonzero delay τ and a mass ratio p> 1 is easiest to understand nearthe hanging-down state θ = 0 (which is a stable periodic state in a large part of the parameterspace of the original system shown in figure 1(a)). In the linearization in θ = 0 of system (5) theE134second equation does not couple back into the first equation such that the linearized equation fory readsy¨+ py¨(t− τ)+β y˙+αy= acos(Ωt). (6)The linear operator corresponding to the homogeneous part of (6) (the left-hand-side) hasinfinitely many eigenvalues and these eigenvalues accumulate to the eigenvalues correspondingto the simpler difference equationy(t) =−py(t− τ), (7)which has infinitely many eigenvalues with real part τ−1 log p. If p> 1 these eigenvalues areunstable regardless of the delay τ . The growth rate even gets larger when the delay is decreased,making the problem practically ill-posed for small delays. This instability carries over to the fullsystem (5) linearized along any trajectory that passes through θ = 0, also leading to infinitelymany unstable eigenvalues for small delays τ > 0; see [4, 10]. The essential spectrum of thetime-τ map of (5) is located outside the unit circle. Hence, we refer to this instability as essential.This instability is, of course, not present in the emulated system (1), (2).Moreover, none of the delay compensation schemes developed for hybrid testing is able to sta-bilize the infinitely many eigenvalues. The delay compensation schemes are based on polynomialextrapolation [3, 9] and have been successfully applied, for example, for the pendulum-MSDsystem with p 1 and for hybrid experiments that are not split at a mass. However, any schemebased on polynomial extrapolation that is consistent (meaning that it is correct for delay τ = 0)must have an essential instability for the split-mass system (5) shown in figure 1(b) if p> 1 andfor delays arbitrarily close to τ [10].A real actuator is not capable of supporting an instability at infinitely many frequencies.Typically, the actuator will be a stiff approximation of the idealization (4), for example,y¨a+ csy˙a+ω2s [ya− y(t− τ)] = F/msfor a large cs > 0 and ωs > 0 where F is the force measured at the pivot. This gives rise to aregularization of the ill-posed problem (5), but still with a large number of strongly unstableeigenvalues for large cs,ωs and small delays, which are still technically impossible to stabilize.3. INTERFACE MATCHING BY NEWTON ITERATIONA consequence of the arguments in the previous section is that for a mass ratio p > 1 it isimpossible to achieve an approximation of the dynamics of the emulated system in figure 1 (a) bya bidirectional real-time coupling as in figure 1 (b). We demonstrate now that it is still possibleto perform a systematic analysis of the dynamics of the emulated system. In order to achieve thiswe break the coupling in one direction, match the output at the interface by a Newton iterationand exploit some fundamental statements from bifurcation theory. For example, in the originalemulated system (1), (2) the hanging-down state θ = 0 loses its dynamical stability in a perioddoubling bifurcation. Standard bifurcation theory states that near this loss of stability a solutionwith a small harmonic amplitude of θ and period 4pi/Ω emerges [7]. Hence, in order to findthe boundary of stability of the hanging-down state it is sufficient to track small period-twosolutions.Figure 2 shows how to break the bidirectional coupling. The actuator is fed with a periodicdemand y˜. In addition, the pendulum experiment is stabilized by a feedback loop with a periodicE134yθsimulation experimentF Fθfeedback θ − θ˜y¨+β y˙+αy=acos(Ωt)+Fθ−θ˜yameasurement:F, θ , yayFyaactuatory˜Figure 2: Sketch of the iterative hybrid experiment. The periodic input y˜ replaces the couplingfrom the simulation to the experiment. Additionally, a classical (proportional-plus-derivative)feedback loop stabilises the parametrically driven pendulum. The system has two additionalperiodic inputs: θ˜ , y˜. A model for this setup is system (8).12 13 14 15 16 1700.511.52x 10−3970 980 990 1000 1010 1020 1030−0.0200.02−0.05 0 0.05−0.1−0.0500.050.1aΩθ − θ˜y˜ytsee (b,c)(b)(c)(a)Figure 3: Results of continuation of period doubling. Panel (a): stability curve in two-parameterplane, panel (b1): control effort in feedback loop along a part of the curve, panel (b2): anticipationof y by y˜.demand signal θ˜ . The unidirectionally coupled system would follow the modely¨=−β y˙−αy+acos(Ωt)+F computer simulationθ¨ =− [1+ y¨a]sinθ −ζ θ˙ −PD[θ − θ˜ ](t− τ) feedback stabilized experimentF(t) = −py¨a− p[θ¨ sinθ + θ˙ 2 cosθ]measured force(8)where PD[x] = kpx+ kd x˙ is a standard proportional-plus-derivative control, ya depends only onF and y˜ but not on y (thus, there is no coupling from the computer simulation to the experiment).The model (8) assumes that this control is applied in the form of a torque at the pivot of thependulum. Due to the feedback loop the unidirectionally coupled system will settle (after atransient) into a locally unique periodic state that can be measured and depends on the givendemands y˜ and θ˜ of period 4pi/Ω. Its measurable outputs are the angular displacement θ [θ˜ , y˜],the motion of the pivot ya[θ˜ , y˜] and the output of the simulation y[θ˜ , y˜,Ω,a]. The followingsystem of nonlinear equations defines the period doubling bifurcation of the hanging-down stateE134of the original system:0 = y[θ˜ , y˜,Ω,a]− ya[θ˜ , y˜] synchronization (9)0 = θ [θ˜ , y˜]− θ˜ control non-invasive (10)r2 =∫ 4pi/Ω0(θ˜(t)−θ0)2 dt period doubling (11)where r is small and θ0 is the average of θ˜ . The variables of this system are the two parametersa and Ω (the excitation in the simulation) and the two periodic control demands θ˜ and y˜, whichcan be expressed by their first two Fourier modes:y˜(t) = y0+ y1 exp(itΩ/2)+ y2 exp(itΩ)θ˜(t) = θ0+θ1 exp(itΩ/2)+θ2 exp(itΩ)where y0,θ0 ∈R, y1, y2, θ1, θ2 ∈C. After a Galerkin projection of (9) and (10) onto the first twoFourier modes one obtains 11 (real-valued) equations for 12 (real-valued) variables (Ω, a, y0, θ0,y1, y2, θ1, θ2). This resulting system defines an implicit curve that can be found by a Newtoniteration embedded into a pseudo-arclength continuation. Each evaluation of the right-hand-sideof (9) and (10) requires one to run the experiments once until the transients have died. Thismakes function evaluations expensive compared to purely numerical bifurcation analysis asdiscussed in [7] and implemented in AUTO [6].Figure 3 shows the results of a proof-of-concept computer simulation using the idealizedactuator model (4) with delay τ = 0.01/ω ≈ 0.07 (10 ms) and the pendulum equation (2)for the pendulum. Figure 3 (a) displays the curve defined by the Galerkin approximation of(9)–(11). Figure 3 (b) shows a typical time profile during the continuation along the perioddoubling curve. Note that the transients, occuring whenever the demands and parameters arechanged, are typically small because demands and parameters are varied only gradually duringthe continuation. The squares along the time axis indicate when the system is considered to havesettled down to a periodic state, giving one evaluation of the right-hand-sides of (9) and (10).The crosses along the time axis mark those function evaluations that correspond to a solutionthat was accepted by the tolerance of the Newton iteration as a solution of (9)–(11). Figure 3 (c)displays in the synchronization plane that y˜ anticipates the output of the simulation y slightly.Importantly, we achieve synchronization without expressly exploiting the knowledge about theactuator model (4).4. CONCLUSION AND FURTHERWORKProof-of-concept computer simulations, including adverse effects such as coupling delay andmeasurement inaccuracies, propose that bifurcation analysis should be possible even for hybridexperiments that are genuinely ill-posed.The incorporation of control-based bifurcation analysis into the hybrid experiment itself iscurrently in preparation. The most pressing problems for the future, apart from experimentalvalidation, are the incorporation of other bifurcations (some have been studied in [12]), ofstrongly nonlinear phenomena (such as homoclinics), and of non-periodic responses.REFERENCES[1] W. Govaerts and Y.A. Kuznetsov. Interactive continuation tools. In B. Krauskopf, H.M.Osinga, and J. Gala´n-Vioque, editors, Numerical Continuation Methods for DynamicalE134Systems: Path following and boundary value problems, pages 51–75. Springer-Verlag,Dordrecht, 2007.[2] A. Blakeborough, M.S. Williams, A.P. Darby, and D.M. Williams. The development ofreal-time substructure testing. Philosophical Transactions of the Royal Society of LondonA, 359:1869–1891, 2001.[3] M.I. Wallace, D.J. Wagg, and S.A. Neild. An adaptive polynomial based forward predictionalgorithm for multi-actuator real-time dynamic substructuring. Proceedings of the RoyalSociety of London A, 461:3807–3826, 2005.[4] Y.N. Kyrychko, K.B. Blyuss, A. Gonzalez-Buelga, S.J. Hogan, and D.J. Wagg. Real-timedynamic substructuring in a coupled oscillator-pendulum system. Proc. Roy. Soc. LondonA, 462:1271–1294, 2006.[5] A. Gonzalez-Buelga, D.J. Wagg, and S.A. Neild. Parametric variation of a coupledpendulum-oscillator system using real-time dynamic substructuring. Structural Controland Health Monitoring, 14(7):991–1012, 2007.[6] E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, andX. Wang. AUTO97, Continuation and bifurcation software for ordinary differential equa-tions. Concordia University, 1998.[7] Yuri A. Kuznetsov. Elements of applied bifurcation theory, volume 112 of Applied Mathe-matical Sciences. Springer-Verlag, New York, third edition, 2004.[8] M.I. Wallace, J. Sieber, S.A. Neild, D.J. Wagg, and B. Krauskopf. Stability analysisof real-time dynamic substructuring using delay differential equation models. Int. J. ofEarthquake Engineering and Structural Dynamics, 34(15):1817–1832, 2005.[9] T. Horiuchi and T. Konno. A new method for compensating actuator delay in real-timehybrid experiments. Philosophical Transactions of the Royal Society A, 359:1893–1909,2001.[10] J. Sieber and B. Krauskopf. Tracking oscillations in the presence of delay-induced essentialinstability. Journal of Sound and Vibrations, (in press), 2008. doi:10.1016/j.jsv.2007.12.016.[11] Jack K. Hale and Sjoerd M. Verduyn Lunel. Introduction to functional-differential equations,volume 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.[12] J. Sieber and B. Krauskopf. Control based bifurcation analysis for experiments. NonlinearDynamics, 51(3):365–377, 2008.E134

Tracking of nonlinear oscillations in hybrid experiments

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Tracking of nonlinear oscillations in hybrid experiments

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