Generalized solvability and optimal control for an integro-differential equation of a hyperbolic type

We consider an integro-differential operator with Volterra type integral term. We provide a priory inequalities in negative norms for certain spaces. Further, using obtained inequalities we prove well-posedness (existence and uniqueness of the (weak) generalized solution) of the corresponding boundary value problem as well as a theorem on optimal control existence.

Feedback Control of Injection Rate of the Injection Molding Machine Based on PID Improved by Unsaturated Integral

In conventional proportional–integral–derivative (PID) control, the integral term occupies a significant amount of controller memory, which prolongs the calculation time. The integral term easily leads to overshooting and oscillation, while the derivative term reduces the controller's anti-interference ability. In view of these problems, a PID expression with a recurrence relationship was derived, and the integral and differential terms of the conventional PID model were improved using an unsaturated integral and passivation differential to achieve a good control effect. Then, the improved PID was applied to the injection speed control of an injection molding machine, which is usually controlled using conventional PID control that featured difficulty in mathematical modeling, a nonlinear relationship between the input and the output, and high system complexity. Taking an injection molding machine as the control object, the transfer function of the injection system was constructed. Then, the improved PID was simulated using Matlab/Simulink. Lastly, the improved PID was verified using experiments. The simulation and the experimental results showed that the control model had a rapid response, no overshoot, and a high precision.

Singularly Perturbed Integro-Differential Equations With Rapidly Oscillating Coefficients and With Rapidly Changing Kernel in the Case of a Multiple Spectrum

The paper investigates a system with rapidly oscillating coefficients and with a rapidly decreasing kernel of the integral operator. Previously, only differential problems of this type were studied in which the integral term was absent. The presence of an integral operator significantly affects the development of an algorithm for asymptotic solutions, for the implementation of which it is necessary to take into account essentially singularities generated by the rapidly decreasing spectral value of the kernel of the integral operator. In addition, resonances can arise in the problem under consideration (i.e., the case can be realized when an integer linear combination of the eigenvalues of the rapidly oscillating coefficient coincides with the points of the spectrum of the limiting operator over the entire considered time interval), as well as the case where the eigenvalue of the rapidly oscillating coefficient coincides with the points spectrum of the limiting operator. This case generates a multiple spectrum of the original singularly perturbed integro-differential system. A similar problem was previously considered in the case of a simple spectrum. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this article.

Equilibrium Optimizer-Based Robust Sliding Mode Control of Magnetic Levitation System

Magnetic Levitation System (MLS) objective is to levitate objects to the desired height without any contact. MLS is highly nonlinear and inherently unstable. Such a system imposes a challenge when designing robust and high-performance controllers. This paper presents the design of a Sliding Mode (SM) controller with an Integral term called SM-I controller to achieve the desired levitation against nonlinearities and uncertainties of the system. The controller parameters are tuned using the Equilibrium Optimizer (EO) algorithm. The Effectiveness of the proposed controller is validated by simulation results. Simulations are performed for servo tracking with and without perturbations in the MLS parameters. The proposed controller is compared with the conventional SM, LQR, and PID controllers to show its superiority. The results prove that the SM-I is more efficient than the other controllers.

Neutron angular flux reconstruction in slab geometry using multigroup discrete ordinates transport models

In this article, we present an application of the coarse-mesh Deterministic Spectral Method (SDM) to generate multigroup angular fluxes in one-dimensional spatial domains using the neutron transport stationary equation, in the formulation of discrete ordinates (SN), considering isotropic scattering source. After obtaining the analytical solution of the SN equations, we replace the integral term of the scattering source in the original neutron transport equation. Thus, we obtain analytically two expressions for angular fluxes in the multigroup formulation, considering the neutron propagation in the positive ( ) and negative ( ) directions, presenting a meaningful reduction in the computational time of simulations of typical neutron shielding problems.

A non-standard numerical scheme for an age-of-infection epidemic model

<p style='text-indent:20px;'>We propose a numerical method for approximating integro-differential equations arising in age-of-infection epidemic models. The method is based on a non-standard finite differences approximation of the integral term appearing in the equation. The study of convergence properties and the analysis of the qualitative behavior of the numerical solution show that it preserves all the basic properties of the continuous model with no restrictive conditions on the step-length <inline-formula><tex-math id="M1">\begin{document}$ h $\end{document}</tex-math></inline-formula> of integration and that it recovers the continuous dynamic as <inline-formula><tex-math id="M2">\begin{document}$ h $\end{document}</tex-math></inline-formula> tends to zero.</p>

Stability of Sampled-Data Control for Lurie Systems with Slope-Restricted Nonlinearities

This paper deals with the stability analysis of aperiodic sampled-data Lurie systems, where the nonlinearity is assumed to be both sector and slope restricted. The proposed method is based on the use of a new class of looped-functionals whose derivative is negative along the trajectories of the continuous-time system. In addition, it contains a generalized Lurie-type function that is quadratic on both the states and the nonlinearity and has a Lurie-Postnikov integral term, which provides some advantages in comparison to simpler candidate functions. On this basis, stability conditions in the form of linear matrix inequalities (LMIs) are formulated. It is shown that the proposed conditions guarantee that the Lurie function is strictly decreasing at the sampling instants, which also implies that the continuous-time trajectories converge asymptotically to the origin. We then formulate some optimization problems for computing themaximal intersampling interval or the maximal sector bounds for which the stability of the sampled-data closed-loop system is guaranteed. A numerical example to illustrate the results is provided.

On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

Robust nonlinear PD controller for ship steering autopilot system based on particle swarm optimization technique

This paper proposes a new approach for robust nonlinear proportional derivative (PD) controller. In this approach a nonlinear function (sigmoid) is added to the conventional proportional integral derivative (PID) controller with filtering for the derivative, in order to improve system response and to reduce the effects of the nonlinearity and uncertainty due to variations of hydrodynamic coefficients of ship with the speed. The gains of nonlinear PD controller are tuned by applying particle swarm optimization (PSO) technique. The simulated results by MATLAB program give satisfactory performance with regard to maximum overshoot, settling time and zero steady state error for step, ramp and proposed trajectory as input to the system. The robustness of the autopilot was checked by changing the plant parameters and adding disturbance to the plant input. The used autopilot is nonlinear PD controller because the gain of integral term by PSO is approximately zero which simplifies the controller construction. The results show that the proposed controller has superior transient response and robustness on the conventional PID designed by using symmetrical optimum criterion with pole assignment technique.

Regularization Method for Singularly Perturbed Integro-Differential Equations with Rapidly Oscillating Coefficients and Rapidly Changing Kernels

In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop an algorithm for the regularization method for such systems and to identify the influence of the integral term on the asymptotic behavior of the solution of the original problem.