Venkatesh Deshmukh

Spectral Collocation-Based Optimization in Parameter Estimation for Nonlinear Time-Varying Dynamical Systems

A constructive optimization algorithm using Chebyshev spectral collocation and quadratic programming is proposed for unknown parameter estimation in nonlinear time-varying dynamic system models to be constructed from available data. The parameters to be estimated are assumed to be identifiable from the data, which also implies that the assumed system models with known parameter values have a unique solution corresponding to every initial condition and parameter set. The nonlinear terms in the dynamic system models are assumed to have a known form, and the models are assumed to be parameter affine. Using an equivalent algebraic description of dynamical systems by Chebyshev spectral collocation and data, a residual quadratic cost is set up, which is a function of unknown parameters only. The minimization of this cost yields the unique solution for the unknown parameters since the models are assumed to have a unique solution for a particular parameter set. An efficient algorithm is presented stepwise and is illustrated using suitable examples. The case of parameter estimation with incomplete or partial data availability is also illustrated with an example.

Propagation of radiation in time-dependent three-dimensional random media

In Ref. [1] (Appendix A) we derived equations governing the frequency and spatial spectrum of radiation propagating in three-dimensional time-dependent random media with randomly varying sound speed c ( x , t). From the spectral equations we determine equations for the energy flux in both the forward and backward directions. We consider media that are spatially homogeneous and isotropic and stationary in time. In order to allow an independence assumption the analysis is restricted to fluctuations that satisfy the conditions τμ ≫ L z /c 0 and τμ ≪ L FS/c 0 where τμ is the characteristic time of the fluctuations, k 0 is the mean radiation wavenumber, L z is the characteristic correlation length of the random fluctuations in the mean propagation direction and L FS is a mean scattering length. We consider various values of γ = (k 0 L z )2/2. When γ ≪ 1 we find the usual radiation transfer equations. When γ ≫ 1, but back-scattering can be neglected, we find the forward-scattering equations. We also consider γ ≫ 1, when back-scattering cannot be neglected. We consider as initial boundary conditions a plane wave and an infinite incoherent source. We present numerical solutions for γ ≪ 1, γ = O (1) and γ ≫ 1 using a simple Gaussian form for the fluctuation correlation function.

Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems

Order reduction of parametrically excited linear and nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and (for nonlinear systems) periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. Two possible order reduction techniques are suggested. In the first approach a constant Guyan-like linear kernel which accounts for both stiffness and inertia is employed with a possible periodically modulated nonlinear part for nonlinear systems. The second method for nonlinear systems reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates. In the process, closed form expressions for “true internal” and “true combination” resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A four degree-of-freedom mass- spring-damper system with periodic stiffness and damping as well as two and five degree-of-freedom inverted pendula with periodic follower forces are used as illustrative examples. The nonlinear-based reduced models are compared with linear-based reduced models in the presence and absence of nonlinear resonances.

Computing Numerical Solutions of Delayed Fractional Differential Equations With Time Varying Coefficients

Fractional differential equations with time varying coefficients and delay are encountered in the analysis of models of metal cutting processes such as milling and drilling with viscoelastic damping elements. Viscoelastic damping is modeled as a fractional derivative. In the present paper, delayed fractional differential equations with bounded time varying coefficients in four different forms are analyzed using series solution and Chebyshev spectral collocation. A fractional differential equation with a known exact solution is then solved by the methodology presented in the paper. The agreement between the two is found to be excellent in terms of point-wise error in the trajectories. Solutions to the described fractional differential equations are computed next in state space and second order forms.

Center Manifold Reduction of Periodic Delay Differential Systems

A technique for center manifold reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. Perturbation expansion converts the nonlinear response problem into solutions of a series of nonhomogenous linear ordinary differential equations (ODEs) with time periodic coefficients. One set of linear nonhomogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. Center manifold reduction on the map is then carried out. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.

Output reversibility in linear discrete-time dynamical systems☆

Output reversibility involves dynamical systems where for every initial condition and the corresponding output there exists another initial condition such that the output generated by this initial condition is a time-reversed image of the original output with the time running forward. Through a series of necessary and sufficient conditions, we characterize output reversibility in linear discrete-time dynamical systems in terms of the geometric symmetry of its eigenvalue set with respect to the unit circle in the complex plane. Furthermore, we establish that output reversibility of a linear continuous-time system implies output reversibility of its discretization. In addition, we present a control framework that allows to alter the system dynamics in such a way that a discrete-time system, otherwise not output reversible, can be made output reversible. Finally, we present numerical examples involving a discretization of a Hamiltonian system that exhibits output reversibility and an example of a controlled system that is rendered output reversible.

Output Reversibility in Linear Discrete-Time Dynamical Systems

Output reversibility involves dynamical systems where for every initial condition and the corresponding output there exists another initial condition such that the output generated by this initial condition is a time-reversed image of the original output with the time running forward. Through a series of necessary and sufficient conditions, we characterize output reversibility in linear single-output discrete-time dynamical systems in terms of the geometric symmetry of its eigenvalue set with respect to the unit circle in the complex plane. Furthermore, we establish that output reversibility of a linear continuous-time system implies output reversibility of its discretization regardless of the sampling rate. Finally, we present a numerical example involving a discretization of a Hamiltonian system that exhibits output reversibility.Copyright

Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations With Time Periodic Coefficients

Stability theory of Nonlinear Delay Differential Algebraic Equations (DDAE) with periodic coefficients is proposed with a geometric interpretation of the evolution of the linearized system. First, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a finite dimensional approximation or a “monodromy matrix” of a compact infinite dimensional operator. The monodromy operator is essentially a map of the Chebyshev coefficients of the state form the delay interval to the next adjacent interval of time. The monodromy matrix is obtained by a similarity transformation of the momodromy matrix of the associated semi-explicit system. The computations are entirely performed in the original system form to avoid cumbersome transformations associated with the semi-explicit system. Next, two computational algorithms are detailed for obtaining solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.Copyright