Stability analysis of a noise control system in a duct by using delay differential equation

The paper aims to study the dynamic behavior of a speed governor for a hydraulic turbine using a mathematical model. The nonlinear mathematical model proposed consists in a system of delay differential equations (DDE) to be compared with already established mathematical models of ordinary differential equations (ODE). A new kind of nonlinearity is introduced as a time delay. The delays can characterize different running conditions of the speed governor. For example, it is considered that spool displacement of hydraulic amplifier might be blocked due to oil impurities in the oil supply system and so the hydraulic amplifier has a time delay in comparison to the time control. Numerical simulations are presented in a comparative manner. A stability analysis of the hydraulic control system is performed, too. Conclusions of the dynamic behavior using the DDE model of a hydraulic turbine speed governor are useful in modeling and controlling hydropower plants.

Modeling by means of differential equations is considered in the paper. Their solutions are constructed on the base of functional relations connecting values of a solution of the equation in different points (infinite or finite set of values). For examples, even, odd and periodical solutions, Vallée-Poussin’s assertion, Lagrange interpolation polynomial, Hermite interpolation polynomial, spline-functions for ordinary differential equations, Asgeirsson’s identity and its generalizations for partial differential equations of hyperbolic type, “mean value” for partial differential equations of elliptic type are considered. Also, if an equation is close to one of considered types then an assertion is to be fulfilled approximately. Some estimations are found for such examples. An application of such relations to investigate some problems of interpolation and extrapolation is demonstrated.

Tuberculosis is a global disease with much significant impact. Well over two million deaths are recorded each year due to Mycobacterium tuberculosis. The work by Colijn et al. (J Theor Biol 247, 765–774, 2007) examined two mathematical models (delay differential equation and network models) that explored the role of the contact structure of the population and found that in declining epidemics, localized outbreaks may occur as a result of contact heterogeneities even in the absence of host or strain variability. In this work, we carry out several qualitative studies (including the use of Sturm sequences for obtaining conditions on bifurcations) on the differential equation model with a time delay in Ref. Colijn et al., J Theor Biol 247, 765–774 (2007), and explicitly obtain threshold conditions for disease eradication as well as existence of bifurcations depending on a critical delay value which characterize the duration of fast latency. Further analysis showed the existence of a transcritical bifurcation when the reproduction number was less than one with the disease-free equilibrium co-existing with just one endemic state that was shown to be stable; the period of fast latency and the reinfection progression rate were shown to be very significant in the dynamics of the disease viz a viz the individuals who progressed to active TB within 5 years of their infection.

The processes of dynamic deformation of shells are actively studied by both domestic and foreign scientists [4; 11; 13; 14]. The work describes the ray method for solving systems of partial differential equations of hyperbolic type. This method consists in constructing the equations of discontinuity transfer along the propagation of perturbations on moving surfaces, as well as in representing the solution in the form of a Taylor power series in the variable distance behind the perturbation front while preserving a sufficient number of terms. The implementation of the method is shown by the example of a system of partial differential equations of hyperbolic type, which describe the process of dynamic deformation of thin cylindrical shells of revolution from elastoviscoplastic materials [10]. An algorithm for constructing a solution up to the required order is given.

Stability analysis of nonsmooth systems, using Lyapunov direct method, is a difficult task. An alternative way of studying the stability in such systems is to employ the concept of Lyapunov exponents. However, two difficulties in the calculation of Lyapunov exponents, i.e. numerical artifacts and low computational efficiency, often prohibit the application of this otherwise powerful method. In this paper, a case study of the stability analysis of a switching contact task control system using the concept of Lyapunov exponents is presented. The goal is to carry out a numerical investigation of the method of nonstandard finite difference discretization, for constructing discrete models of differential equations that describe the Lyapunov exponents for such a nonsmooth system. It is shown that, as compared with the standard fourth-order Runge-Kutta method, the nonstandard finite difference discretization scheme provides numerically stable results with a larger critical integration step-size and less computation time. Therefore, from numerical stability and computational efficiency viewpoints, the nonstandard finite difference discretization method is meritorious and should be given consideration when stability analysis of systems using Lyapunov exponents is of concern.

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

Model-based strategies for sensor fault accommodation in uncertain dynamic processes with multi-rate sampled measurements

Applications of bifurcation theory have been employed in all engineering branches, especially in control. In this paper we analyse one control application to delay differential equations based on mentioned bifurcation theory. The time response of a first order delay model with saturation effects has been studied. A stability parameters map, showing the boundaries which separate zones with different qualitative behaviour, has been obtained by analysing the characteristic transcendental equation. Different cases for initial conditions have been employed in our analysis.

In a rectangular domain, we consider a boundary value problem periodic in one variable for a system of partial differential equations of hyperbolic type. Introducing a new unknown function, this problem is reduced to an equivalent boundary value problem for an ordinary differential equation with an integral condition. Based on the parametrization method, new approaches to finding an approximate solution to an equivalent problem are proposed and its convergence is proved. This made it possible to establish conditions for the existence of a unique solution of a semiperiodic boundary value problem for a system of second-order hyperbolic equations.

Partial differential equations of hyperbolic type when considered with mixed Dirichlet/Neumann constraints as well as nonlocal conservation conditions model many physical phenomena. The prime motivation of the current work is to apply the recently developed meshfree method to such differential equations. The scheme is built on series expansion of the solution via proper base functions akin to the Galerkin approach. In many cases, the simple polynomials are adequate to convert the hyperbolic partial differential equation and boundary conditions of nonlocal kind into easily treatable algebraic equations concerning the coefficients of the series. If the sought solutions are polynomials of any degree, then the method has the ability of resolving the equations in an exact manner. The validity, applicability, accuracy and performance of the method are illustrated on some well-analyzed hyperbolic equations available in the open literature.

In many areas, researchers might think that a differential equation model is required, but one might be forced to use an approximate difference equation model if data is only available at discrete points in time. In this paper, a detailed comparison is given of the behavior of continuous and discrete models for two representative time-delay models, namely a model for HIV and an extended logistic growth model. For each model, there are seven different time-delay versions because there are seven different positions to include time delays. For the seven different time-delay versions of each model, proofs are given of necessary and sufficient conditions for the existence and stability of equilibrium points and for the existence of Andronov–Hopf bifurcations in the differential equations and Neimark–Sacker bifurcations in the difference equations. We show that only five of the seven time-delay versions have bifurcations and that all bifurcation versions have supercritical limit cycles with one having a repelling cycle and four having attracting cycles. Numerical simulations are used to illustrate the analytical results and to show that critical times for Neimark–Sacker bifurcations are less than critical times for Andronov–Hopf bifurcations but converge to them as the time step of the discretization tends to zero.

Fuzzy inference is a method to describe nonlinear input-output relationships using fuzzy if-then rules. Continuous values of the inputs and outputs are converted into granules by fuzzy sets, and each granule is labeled with a symbol. Fuzzy inference has a multigranular architecture consisting of continuous values and symbols, and this architecture has worked well to incorporate experts' know-how into fuzzy controls. One of the important problems of fuzzy control is to guarantee stability of the fuzzy control system. The authors have applied Petri nets to the stability analysis of the fuzzy control system. A theory of asymptotic stability has been derived for the symbolic representation of the control system. The paper presents a new method to bridge between the stability analysis on the symbolic level and the actual behavior of the control system on the numerical level. The new method uses a generalized fuzzy Petri net model and its neural network representation. The paper introduces a guideline for designing a fuzzy control system which guarantees the validity of the stability analysis on the symbolic representation of the control system.

Based upon the Piece-wise Affine (PWA) model of closed-loop Model Predictive Control (MPC) system and Lyapunov functions, stability analysis of the closed loop MPC system is studied. By trying to find piece-wise quadratic Lyapunov function over the state’s regions, which is cast as an LMI problem, the stability of closed-loop MPC system is analyzed. With further convex partition of the state’s regions, and thus increasing the possibility to find the piece-wise quadratic Lyapunov functions, the conservatism of the stability analysis is reduced. At the end of the paper, simulations for a vibration control of elevator’s mechanical system are made.

For flexible robots with rotational joints it has been shown previously by Luo (1993), that direct strain feedback can damp out vibrations very satisfactorily. In this paper, a simple sensor-based output feedback control law, called shear force feedback, is newly proposed to control vibrations arising from structural flexibility of robots of Cartesian or SCARA types. Closed-loop exponential stability of such shear force feedback system is proved. Experimental results on set point control and trajectory tracking control are reported. It is found that the simple PI+shear force feedback can yield good performance for both robot motion and vibration suppression.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Time delay is prejudicial to power system stability and generally worsens the performance of control system. In this paper, the approaches for stability analysis and stabilization of a power system consisting of time-varying delay are established. The paper starts with modeling of the power system according to time-delay differential algebraic equations. An Integral Quadratic Constraints (IQC) method is constructed to guarantee the stability of proposed feedback controller. The frequency-domain IQC and time-domain dissipation inequality are effectively combined to perform stability analysis and stabilization of power system considering time delays, without complex numerical computation. A 2-area 4-machine system, an IEEE ten-machine thirty nine-bus system and a WECC 179-bus system are employed to support the stability criteria and IQC method. The simulation results indicate that the combined methods effectively damp the oscillation and improve the performance.