On the dynamic reconstruction of the input of a control system

We consider the problem of reconstructing the right-hand side of a dynamical system subjected to external disturbances. Under the assumption that the states of the system are measured with some error, we indicate an algorithm for solving the problem on the basis of the method of auxiliary positional control models. The algorithm is stable under noises and numerical errors.     

An algorithm for dynamic reconstruction of input disturbances from observations of some of the coordinates

The dynamic reconstruction of input disturbances and unobserved state coordinates is considered. An algorithm based on the dynamic inversion theory is proposed, which is robust to observational and computational errors.     

On reconstruction of an input of a parabolic equation on an infinite time interval

For a parabolic equation we consider the problem of the dynamic reconstruction of inputs from measurements of phase coordinates on an infinite time interval. We propose an algorithm based on constructions of the dynamic inversion theory. The algorithm is stable with respect to informational noises and computational errors.     

An algorithm for dynamic reconstruction of a disturbance in a linear system

We study the problem of dynamic reconstruction of an unknown disturbance in a linear system whose states are measured with some error. We propose a solution algorithmfor this problem under the assumption that the observed time interval is sufficiently large. The algorithm is based on the method of auxiliary positional control models.     

On the analysis of a dynamic evolutionary algorithm

Evolutionary algorithms are applied as problem-independent optimization algorithms. They are quite efficient in many situations. However, it is difficult to analyze even the behavior of simple variants of evolutionary algorithms like the (1+1) EA on rather simple functions. Nevertheless, only the analysis of the expected run time and the success probability within a given number of steps can guide the choice of the free parameters of the algorithms. Here static (1+1) EAs with a fixed mutation probability are compared with dynamic (1+1) EAs with a simple schedule for the variation of the mutation probability. The dynamic variant is first analyzed for functions typically chosen as example-functions for evolutionary algorithms. Afterwards, it is shown that it can be essential to choose the suitable variant of the (1+1) EA. More precisely, functions are presented where each static (1+1) EA has exponential expected run time while the dynamic variant has polynomial expected run time. For other functions it is shown that the dynamic (1+1) EA has exponential expected run time while a static (1+1) EA with a good choice of the mutation probability has polynomial run time with overwhelming probability.     

An algorithm for dynamic reconstruction of the right-hand side of a second-order equation with distributed parameters

The dynamic reconstruction of the right-hand side of a second-order differential equation is considered. A solution algorithm is proposed that is robust to information noise and computational errors. The algorithm is constructed using dynamic inversion theory.     

Some algorithms for the dynamic reconstruction of inputs

For the weight \(v_k \left( x \right) = \prod _{\alpha \in \mathbb{R}_ + } \left| {\left( {\alpha ,x} \right)} \right|^{2k\left( \alpha \right)}\) defined by a positive subsystem R ₊ of a finite root system R ? ? ^d and by a function k(α): R → ?₊ invariant under the reflection group generated by R, a sharp Jackson inequality in L ₂(? ^d ) is proved.     

On a reconstruction algorithm for the trajectory and control in a delay system

We discuss a problem of the dynamic reconstruction of unmeasured coordinates of the phase vector and unknown controls in nonlinear vector equations with delay. A regularizing algorithm is proposed for the reconstruction of both controls and unmeasured coordinates simultaneously with the processes. The algorithm is stable with respect to information noises and computational errors.     

Fast and accurate algorithm for cavities reconstruction in an elasticity problem

The aim of this work is to reconstruct the location and geometry of a cavity embedded in a linear isotropic material Ω via an exterior boundary measurement of the displacement field. The considered problem is governed by the linear elasticity system. This inverse problem of geometry reconstruction (ie, location and shape) is formulated as a topology optimization one and solved by minimizing a Kohn‐Vogelius type functional with the help of the topological sensitivity method. Some numerical results are presented using a noniterative geometric algorithm.     

On an algorithm for calculating diffraction integrals

An algorithm for calculating integrals of rapidly oscillating functions given on a smooth two-dimensional surface is proposed. The surface is approximated by a collection of flat triangles with the values of the integrand known at their vertices. These values are used as reference ones to extend the function to other points of a triangle. The integral of the extended function over the surface of a triangle is calculated exactly. The desired value of the full diffraction integral is determined as the sum of the integrals calculated over the surfaces of all triangles. The resulting formulas for integral calculation involve singularities (indeterminate forms). Much attention is given to representations of these formulas in such a way that the indeterminate forms are automatically evaluated. Numerical results are presented.     

Optimal input system identification for nonlinear dynamic systems

The estimation accuracy for nonlinear dynamic system identification is known to be maximized by the use of optimal inputs. Few examples of the design of optimal inputs for nonlinear dynamic systems are given in the literature, however. The performance criterion is selected such that the sensitivity of the measured state variables to the unknown parameters is maximized. The application of Pontryagin's maximum principle yields a nonlinear two-point boundary-value problem. In this paper, the boundary-value problem for a simple nonlinear example is solved using two different methods, the method of quasilinearization and the Newton-Raphson method. The estimation accuracy is discussed in terms of the Cramer-Rao lower bound.     

On the convergence of an algorithm for discreteL p approximation

Summary The convergence properties of an algorithm for discreteL _p approximation (1p<2) that has been considered by several authors are studied. In particular, it is shown that for 1<p<2 the method converges (with a suitably close starting value) to the best approximation at a geometric rate with asymptotic convergence constant 2-p. A similar result holds forp=1 if the best approximation is unique. However, in this case the convergence constant depends on the function to be approximated.     

Sequential generation ofD-optimal input designs for linear dynamic systems

This paper considers the problem of sequentially generating test signals for parameter estimation in linear, single-input, single-output, discrete-time systems using a frequency-domain approach based on the theory of optimal experiments in regression analysis. The input signals are power constrained and are optimal in the sense that system information is maximized, where the criterion employed is the determinant of the Fisher information matrix (D-optimality). A class of algorithms is investigated, each member of which generates a sequence of input designs converging to aD-optimum. A number of these algorithms are compared computationally.     

On an unknown algorithm for computing square roots

A relatively unknown algorithm for computing square roots is presented. In a survey of more than 50 teachers of mathematics, not one of them recalled having ever encountered it before. The algorithm is first illustrated and then proved.     

On the problem of dynamic control reconstruction from measurements of part of the coordinates

We consider a control dynamical system described by a vector differential equation. We construct a real-time finite-step algorithm that, for the case of small measurement errors, reconstructs a control implementation close to the actual one in the sense of mean-square deviation. The algorithm is represented by a positional control process for an auxiliary finite-dimensional model.     

On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process

Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme.     

On the dynamic solution of an operator inverse problem

We exhibit an algorithm for solving an operator inverse problem that is stable under informational noise and computational errors. The algorithm is based on the constructions of the theory of positional control and Tikhonov's regularization method (the smoothing functional method). Bibliography: 15 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 28–35.