M. Tadi

Inverse heat conduction based on boundary measurement

This paper is concerned with the evaluation of the diffusion coefficient based on the measurement obtained at the boundary. We consider the problem of recovering the diffusion coefficient of a rod that is a function of space. The approach is based on allowing the unknown function to depend on time. With appropriate measurement data the unknown system parameters are guided from an arbitrary initial condition to their true value at a final time. An explicit equation describing the time evolution of the parameter is obtained by minimizing the error along the trajectory. The method leads to an iterative algorithm which is described in detail. In order to present the method in detail we first consider the problem of recovering a scalar heat generation which depends on the local temperature along the rod. Numerical results with the method indicate that close estimates of the unknown function can be obtained. Convergence of the method is also studied for a particular problem in inverse heat conduction.

State-Dependent Riccati Equation for Control of Aeroelastic Flutter

Feedback control design for suppression of flutter in an aeroelastic system is studied. Feedback control laws based on the state-dependent Riccati equation are derived. Two control laws are studied: one formulation uses an observer to estimate the states after which it uses the estimated states to generate feedback law; the other formulation uses partial observation to directly generate the control law.

An inversion method for parabolic equations based on quasireversibility

Abstract This paper is concerned with a new method to solve a linearized inverse problem for one-dimensional parabolic equations. The inverse problem seeks to recover the subsurface absorption coefficient function based on the measurements obtained at the boundary. The method considers a temporal interval during which time dependent measurements are provided. It linearizes the working equation around the system response for a background medium. It is then possible to relate the inverse problem of interest to an ill-posed boundary value problem for a differential-integral equation, whose solution is obtained by the method of quasireversibility. This approach leads to an iterative method. A number of numerical results are presented which indicate that a close estimate of the unknown function can be obtained based on the boundary measurements only.

Focused bulk ultrasonic waves generated by ring‐shaped laser illumination and application to flaw detection

Focused bulk ultrasonic waves have been generated in aluminum plates by surface irradiation with ring‐shaped laser light. The waves are detected by a piezoelectric transducer. Compression and shear peak amplitudes drop quickly when the detector is moved away from the epicenter. This shows that strong focusing exists at the epicenter as the result of constructive interference of the waves generated by different parts of the ring. The focusing persists when the radius of the laser light is scanned over a large range, indicating that the elastic disturbance concentrates in depth along the ring’s central axis. Numerical simulations are presented for comparison. The ‘‘pencil‐like’’ acoustic wave structure is used to observe a sample plate with an artificial flaw. Strong new features including compress‐shear mode conversion at the site of the flaw are observed. These features are used to locate the flaw within the sample.

An iterative method for the solution of ill-posed parabolic systems

This note is concerned with the identification of the unknown initial condition for a parabolic system. It introduces an iterative algorithm that can be used to recover the initial condition. The algorithm assumes a function for the unknown initial condition and obtains a background field. It then obtains the error dynamics. The error dynamics is also ill-posed in the sense that the initial condition is unknown. Using the measurements obtained at the boundaries, the algorithm introduces two formulations for the error dynamics. These two formulations share the same initial condition. By equating the responses of these two formulations it is then possible to obtain an equation for the unknown initial condition. A number of numerical examples are also used to study the performance of the algorithm.

Optimal infinite-dimensional estimator and compensator for a Timoshenko beam

Abstract This paper is concerned with computational algorithms to obtain convergent functional estimator gains for a Timoshenko slewing beam. It is shown that the states of the system can be optimally estimated and such estimates can be used to construct a compensator for the system. Effectiveness of the compensator is also compared to the case when the full state is available for feedback.

Explicit method for inverse wave scattering in solids

This paper is concerned with an inverse problem for a one-dimensional rod through a new method which treats the unknown parameters as time dependent. The method was originally developed for dynamical systems involving ordinary differential equations. With appropriate measurement data the unknown system parameters are guided from an arbitrary initial condition to their true value at a final time. An explicit equation describing the time evolution of the parameter is obtained by minimizing the error along the trajectory. The method leads to an iterative algorithm which is described in detail. A one-dimensional elastic rod whose density is a function of space is considered. Numerical results with the method indicate that accurate estimates of the unknown function can be obtained even in the presence of noise in the data.

Computational algorithm for controlling a Timoshenko beam

Abstract This paper is concerned with computational algorithms for LQR control of a Timoshenko slewing beam problem. The algorithm make use of a conforming finite element scheme that is shown to produce convergent control laws. It is shown that the closed loop feedback can effectively control the vibrations due to an initial displacement. It is also shown that when the system parameters are such that an Euler-Bernoulli model is adequate to describe the dynamics of the beam, the functional gains using the full Timoshenko model are qualitatively similar to the ones using an Euler-Bernoulli model.

Explicit Method for Parameter Identification

Parameter identie cation of dynamical systems through a new method that treats the unknown parameters as timedependentisreported.Withappropriateobservationaldata,theunknown systemparametersareguided from an arbitrary initialconditionto theirtruevalueat a e nal time. An explicit equationdescribing thetimeevolutionof theparametersisobtainedbyminimizing theerroralongthetrajectory.Themethodleadstoaniterativealgorithm, which is described in detail. Numerical results with the method indicate that accurate estimates of the unknown parameters can be obtained. I. Introduction T HIS paper presents a new method for parameter identie cation of dynamical systems. This activity falls in the general domain of inverse problems where the interest is to obtain the unknown system parameters from the given output measurement of the system. In many applications there is a nonlinear relationship between the measured output and the desired system unknowns. The inverse problemis further complicated by the fact that the measured data will contain noise and are often insufe cient to uniquely specify a solution. A number of techniques have been developed to treat such problems. A brief summary of results for systems described by ordinary differential equations can be found in, for example, Refs. 1 and 2. These approaches include the method of least squares, stochastic least squares, maximum likelihood, and a number of other variations. These schemes have also been applied to dynamical systems governed by partial differential equations. 3,4 A generalized class of linear inversion methods has been applied to a number of physical systems involving replacement of the nonlinear relationship between the observed data and the unknowns with a linear approximation. A comprehensive review of this approach can be found. 5,6 Of particular interest are direct inversion schemes for continuous systemsthatarebasedonregularization. 7 Inthispaper,we approach the parameter identie cation problem for systems governed by ordinary differential equations in a way that permits a similar treatment of the problem. We then use regularization to stabilize the inversion. The method leads to an explicit differential equation for the unknown parameters, which is used in an iterative algorithm. In this paper, we do not address the question of system identie ability . 8 We present a method that can be used to estimate the parameters for systems that are identie able. Thepaperisorganizedasfollow.InSec.II,wepresentthegeneral algorithm along with a number of numerical examples. Section III considers an extension of the method to a broader class of systems, including further numerical results, and Sec. IV is devoted to conclusions.

An inverse problem for Helmholtz equation

This article is concerned with subsurface material identification for the 2-D Helmholtz equation. The algorithm is iterative in nature. It assumes an initial guess for the unknown function and obtains corrections to the guessed value. It linearizes the otherwise nonlinear problem around the background field. The background field is the field variable generated using the guessed value of the unknown function at each iteration. Numerical results indicate that the algorithm can recover a close estimate of the unknown function based on the measurements collected at the boundary.