Fast generation of stability charts for time-delay systems using continuation of characteristic roots

Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method significantly reduces the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.Comment: 12 pages, 6 figure

Nonlinear Dynamics of Heat-Exchanger Tubes Under Crossflow: A Time-Delay Approach

Fluid-conveying heat-exchanger tubes in nuclear power plants are subjected to a secondary cross-flow to facilitate heat exchange. Beyond a critical value of the secondary flow velocity, the tube loses stability and vibrates with large amplitude. The equation governing the dynamics of a heat-exchanger tube is a delay differential equation (DDE). In all the earlier studies, only the stability boundaries in the parametric space of mass-damping parameter and reduced flow-velocity were reported. In this work using Galerkin approximations, the spectrum (characteristic roots) of the DDE is also obtained. The rightmost characteristic root, whose real part represents the damping in the heat-exchanger tube is included in the stability chart for the first time. The highest damping is found to be present in localized areas of the stability charts, which are close to the stability boundaries. These charts can be used to determine the optimal cross-flow velocities for operating the system for achieving maximum damping. Next, the interaction between the tube and the surrounding cladding at the baffle-plate makes it vital to determine the optimal design parameters for the baffle plates. The linear stability of a heat-exchanger tube modeled as a single-span Euler-Bernoulli cantilever beam subjected to cross-flow is studied with two parameters: (i) varying stiffness of the baffle-cladding at the free end and (ii) varying flow velocity. The partial delay differential equation governing the dynamics of the continuous system is discretized to a set of finite, nonlinear DDEs through a Galerkin method in which a single mode is considered. Unstable regions in the parametric space of cladding stiffness and flow velocity are identified, along with the magnitude of damping in the stable region. This information can be used to determine the design cladding stiffness to achieve maximum damping at a known operational flow velocity. Moreover, the system is found to lose stability by Hopf bifurcation and the method of multiple scales is used to analyze its post-instability behavior. Stable and unstable limit cycles are observed for different values of the linear component of the dimensionless cladding stiffness. An optimal range for the linear cladding stiffness is recommended where tube vibrations would either diminish to zero or assume a relatively low amplitude associated with a stable limit cycle. Furthermore, heat-exchanger tubes undergo thermal expansion, and are consequently subject to thermal loads acting along the axial direction, apart from design-induced external tensile loads. Nonlinear vibrations of a heat-exchanger tube modeled as a simply-supported EulerBernoulli beam under axial load and cross-flow have been studied. The fixed points (zero and buckled equilibria) of the nonlinear DDE are found, and their linear stability is analyzed. The stability of the DDE is investigated in the parametric space of fluid velocity and axial load. The method of multiple scales is used to study the post-instability behavior for both zero and buckled equilibria. Multiple limit-cycles coexist in the parametric space, which has implications on the fatigue life calculations of the heat-exchanger tubes

Complete dimensional collapse in the continuum limit of a delayed SEIQR network model with separable distributed infectivity

We take up a recently proposed compartmental SEIQR model with delays, ignore loss of immunity in the context of a fast pandemic, extend the model to a network structured on infectivity, and consider the continuum limit of the same with a simple separable interaction model for the infectivities $\beta$. Numerical simulations show that the evolving dynamics of the network is effectively captured by a single scalar function of time, regardless of the distribution of $\beta$ in the population. The continuum limit of the network model allows a simple derivation of the simpler model, which is a single scalar delay differential equation (DDE), wherein the variation in $\beta$ appears through an integral closely related to the moment generating function of $u=\sqrt{\beta}$. If the first few moments of $u$ exist, the governing DDE can be expanded in a series that shows a direct correspondence with the original compartmental DDE with a single $\beta$. Even otherwise, the new scalar DDE can be solved using either numerical integration over $u$ at each time step, or with the analytical integral if available in some useful form. Our work provides a new academic example of complete dimensional collapse, ties up an underlying continuum model for a pandemic with a simpler-seeming compartmental model, and will hopefully lead to new analysis of continuum models for epidemics

New approximations, and policy implications, from a delayed dynamic model of a fast pandemic

We study an SEIQR (Susceptible-Exposed-Infectious-Quarantined-Recovered) model for an infectious disease, with time delays for latency and an asymptomatic phase. For fast pandemics where nobody has prior immunity and everyone has immunity after recovery, the SEIQR model decouples into two nonlinear delay differential equations (DDEs) with five parameters. One parameter is set to unity by scaling time. The subcase of perfect quarantining and zero self-recovery before quarantine, with two free parameters, is examined first. The method of multiple scales yields a hyperbolic tangent solution; and a long-wave approximation yields a first order ordinary differential equation (ODE). With imperfect quarantining and nonzero self-recovery, the long-wave approximation is a second order ODE. These three approximations each capture the full outbreak, from infinitesimal initiation to final saturation. Low-dimensional dynamics in the DDEs is demonstrated using a six state non-delayed reduced order model obtained by Galerkin projection. Numerical solutions from the reduced order model match the DDE over a range of parameter choices and initial conditions. Finally, stability analysis and numerics show how correctly executed time-varying social distancing, within the present model, can cut the number of affected people by almost half. Alternatively, faster detection followed by near-certain quarantining can potentially be even more effective

Pole Placement and Reduced-Order Modelling for Time-Delayed Systems Using Galerkin Approximations

The dynamics of time-delayed systems (TDS) are governed by delay differential equa- tions (DDEs), which are infinite dimensional and pose computational challenges. The Galerkin approximation method is one of several techniques to obtain the spectrum of DDEs for stability and stabilization studies. In the literature, Galerkin approximations for DDEs have primarily dealt with second-order TDS (second-order Galerkin method), and the for- mulations have resulted in spurious roots, i.e., roots that are not among the characteristic roots of the DDE. Although these spurious roots do not affect stability studies, they never- theless add to the complexity and computation time for control and reduced-order modelling studies of DDEs. A refined mathematical model, called the first-order Galerkin method, is proposed to avoid spurious roots, and the subtle differences between the two formulations (second-order and first-order Galerkin methods) are highlighted with examples. For embedding the boundary conditions in the first-order Galerkin method, a new pseudoinverse-based technique is developed. This method not only gives the exact location of the rightmost root but also, on average, has a higher number of converged roots when compared to the existing pseudospectral differencing method. The proposed method is combined with an optimization framework to develop a pole-placement technique for DDEs to design closed-loop feedback gains that stabilize TDS. A rotary inverted pendulum system apparatus with inherent sensing delays as well as deliberately introduced time delays is used to experimentally validate the Galerkin approximation-based optimization framework for the pole placement of DDEs. Optimization-based techniques cannot always place the rightmost root at the desired location; also, one has no control over the placement of the next set of rightmost roots. However, one has the precise location of the rightmost root. To overcome this, a pole- placement technique for second-order TDS is proposed, which combines the strengths of the method of receptances and an optimization-based strategy. When the method of receptances provides an unsatisfactory solution, particle swarm optimization is used to improve the location of the rightmost pole. The proposed approach is demonstrated with numerical studies and is validated experimentally using a 3D hovercraft apparatus. The Galerkin approximation method contains both converged and unconverged roots of the DDE. By using only the information about the converged roots and applying the eigenvalue decomposition, one obtains an r-dimensional reduced-order model (ROM) of the DDE. To analyze the dynamics of DDEs, we first choose an appropriate value for r; we then select the minimum value of the order of the Galerkin approximation method system at which at least r roots converge. By judiciously selecting r, solutions of the ROM and the original DDE are found to match closely. Finally, an r-dimensional ROM of a 3D hovercraft apparatus in the presence of delay is validated experimentally

Single-shooting homotopy method for parameter identification in dynamical systems

An algorithm for identifying parameters in dynamical systems is developed in this work using homotopy transformations and the single-shooting method. The equations governing the dynamics of the mathematical model are augmented with observer-like homotopy terms that smooth the objective function. As a result, premature convergence to a local minimum is avoided and the obtained parameter estimates are globally optimal. Numerical examples are presented to demonstrate the application of the proposed approach to chaotic systems

Performance limit for base-excited energy harvesting, and comparison with experiments

We consider the theoretical maximum extractable average power from an energy harvesting device attached to a vibrating table which provides a unidirectional displacement $A\sin(\omega t)$. The total mass of moving components in the device is $m$. The device is assembled in a container of dimension $L$, limiting the displacements and deformations of components within. The masses in the device may be interconnected in arbitrary ways. The maximum extractable average power is bounded by $\frac{mLA\omega^3}{\pi}$, for motions in 1, 2, or 3 dimensions; with both rectilinear and rotary motions as special cases; and with either single or multiple degrees of freedom. The limiting displacement profile of the moving masses for extracting maximum power is discontinuous, and not physically realizable. But smooth approximations can be nearly as good: with $15$ terms in a Fourier approximation, the upper limit is $99$\% of the theoretical maximum. Purely sinusoidal solutions are limited to $\frac{\pi}{4}$ times the theoretical maximum. For both single-degree-of-freedom linear resonant devices and nonresonant whirling devices where the energy extraction mimics a linear torsional damper, the maximum average power output is $\frac{mLA\omega^3}{4}$. Thirty-six experimental energy harvesting devices in the literature are found to extract power amounts ranging from $0.0036$\% to $29$\% of the theoretical maximum. Of these thirty-six, twenty achieve less than 2\% and three achieve more than 20\%. We suggest, as a figure of merit, that energy extraction above $\frac{0.2 mLA\omega^3}{\pi}$ may be considered excellent, and extraction above $\frac{0.3 mLA\omega^3}{\pi}$ may be considered challenging

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs

Adaptive sparse Galerkin methods for vibrating continuous structures

Adaptive reduced-order methods are explored for simulating continuous vibrating structures. The Galerkin method is used to convert the governing partial differential equation (PDE)into a finite-dimensional system of ordinary differential equations (ODEs) whose solution approximates that of the original PDE. Sparse projections of the approximate ODE solution are then found at each integration time step by applying either the least absolute shrinkage and selection operator (lasso) or the optimal subset selection method. We apply the two projection schemes to the simulation of a vibrating Euler–Bernoulli beam subjected to nonlinear unilateral and bilateral spring forces. The subset selection approach is found to be superior for this application, as it generates a solution with similar sparsity but substantially lower error than the lasso