Discrete-Time Systems

We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the density of which, at a fixed instant in time, we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the volatility density. An expansion of the bias and a bound on the variance are derived.

This chapter addresses synchromodal planning at operational level from the perspective of a logistics service provider and studies an optimisation problem with simultaneous vehicle routing and container-to-mode assignment, having uncertain data. It is assumed that the release times of the containers belong to an uncertainty interval, and no further statistical information is available. This problem belongs to the fourth quadrant of Fig. 1.4 . The container routing problem is modelled as a mixed integer program with explicit time variables and lateness penalties. A robust formulation is then proposed to eliminate the uncertain parameters from the objective function and constraints. By solving the new model exactly, with the aid of an optimisation solver, robust solutions are obtained corresponding to transportation plans which remain feasible for any realisation of the release times within the pre-specified uncertainty interval. In order to introduce some flexibility in the transportation plan, the continuous time variables are modelled as affine functions of the uncertain parameters. The resulting two-stage decision model is tested for a small-sized instance in both situations, with high and low lateness penalties.

SummaryThis paper investigates the stochastic synchronization problem of delayed multiagent networks with intermittent communications. Two kinds of impulsive effects are taken into account, ie, impulsive controller (positive impulsive effect) and impulsive disturbance (negative impulsive effect). Impulsive controller allows the synchronization to be realized and requires only state information exchange at discrete time instants such that the communication cost of bandwidth is reduced. Meanwhile, impulsive disturbance is inevitable in most of practical systems and therefore is taken into consideration at discrete time instants. Sufficient conditions for synchronization are given in terms of the graph topology, the control coupling gains, and the individual agent dynamics parameters, which indicates that synchronization can be realized if the impulsive effects coefficients and communication rate are suitably selected. Simulation results verify the effectiveness of the proposed synchronization protocol.

Stochastic perturbations can steer the response of a nonlinear system to specific regions of the system state space, including equilibrium points, periodic solutions, and aperiodic (including chaotic) solutions. This study constructs a discrete partial control scheme to confine the response trajectories of a forced Duffing oscillator, showing transient chaos, within a chaotic attractor, thereby avoiding undesired behavior. The system is analyzed in the presence of white noise, using a minimally bounded control input. The primary motivation is to evaluate the viability of applying bounded control inputs at discrete time instants over intervals differing from the forcing period and to examine the required control input magnitude for response confinement. Results demonstrate that the system can be effectively controlled with relatively small magnitude control inputs applied at discrete time instants. A careful selection of time steps is crucial to ensure that the response trajectory remains within the desired region. The scheme's formulation and the associated algorithm are independent of the system dimension.

In the present chapter, we evaluate a formula of population size due to epidemics model problem for communicable diseases in which the daily contact rate is supposed to be varied with population size N (t) that is large so that it is considered as a continuous variable of time t. Then, we obtain some inequalities of above analytic formula of population size. We present an analysis to discuss the epidemics. These results may also be useful to plot some graphs on population dynamics of the system.

In the present paper, we evaluate a formula of population size due to epidemics model problem for communicable diseases, in which the daily contact rate is supposed to be varied with population size N(t) that is large, so that it is considered as a continuous variable of time (t). Then, we obtain some inequalities of above analytic formula of population size that are useful to plot some graphs on population dynamics of the system. Key words: Population size, epidemics model, hypergeometric function, inequalities, communicable diseases.

Harmonic Balance Method for non-linear periodic flows is presented in this paper. As- sumption of temporal periodicity can be used to convert harmonic transient equations into a set of coupled steady state equations. Solution of coupled equations yields flow fields at discrete instants of time throughout a representative harmonic period. The method uses second-order accurate, polyhedral Finite Volume Method. It is developed for a general transport equation and generalised to coupled non-linear equation sets, including incom- pressible Navier-Stokes equations and is implemented in OpenFOAM. Verification of the model is carried out by comparing the results with transient simulations. Preliminary re- sults for weakly and strongly harmonic incompressible flow around oscillating airfoil are presented.

The presented effort is the extract of a composite work. Primarily to model a dc servo-system within the available framework through observed data. The observed data of the hardware are classified as input and output samples measured at discrete instants of time and collected as an array of data for finite duration. Several input-output data sets are collected and processed to find out the model of the dc servo motor through System Identification Tool Box of MATLAB, a Graphical User Interface (GUI) based approach. The fitness of identified model is checked every time new data set is applied to choose the best fitted model. Secondarily to demonstrate in steps the ways of design and implementation of Internal Model Principle (IMP) & Finite Dimensional Repetitive Control (FDRC) based Proportional Integral Derivative (PID) controller for the plant. A similar test has been made with the system using conventional PID control. Finally, a valid comparison is carried out to show advantage, disadvantage or limitation of applied control schemes.

The problem of H∞ filtering design is studied for non-linear sampled-data systems using the Takagi–Sugeno (T–S) fuzzy model approach. The sampled-data filtering is to estimate the states of a continuous-time system using only sampled measurements at discrete instants of time. Traditionally, the sufficient conditions for the existence of such an H∞ filter are characterised in terms of the solution of a differential Hamilton–Jacobi inequality with jumps, which is equivalent to solving the partial differential inequality with jumps. In general, there is no analytical solution for this non-linear partial differential inequality with jumps. First, in this study, the T–S fuzzy model is proposed to represent a class of non-linear sampled-data systems. Next, by using the T–S fuzzy model, the H∞ fuzzy filtering design problem for non-linear sampled-data system is characterised in terms of a linear matrix inequality (LMI) problem. Hence, the H∞ fuzzy filter of non-linear sampled-data systems can be given via solving LMIs instead of solving a differential Hamilton–Jacobi inequality with jumps. To illustrate the results, a numerical example is included.

The creep of homogenous and hybrid composite beams of an irregular laminar fibrous structure is investigated. The beams consist of thin walls and flanges (load-carrying layers). The walls may be reinforced longitudinally or crosswise in the plane, and the load-carrying layers are reinforced in the longitudinal direction. The mechanical behavior of phase materials is described by the Rabotnov nonlinear hereditary theory of creep taking into account their possible different resistance to tension and compression. On the basis of hypotheses of the Timoshenko theory, with using the method of time steps, a problem is formulated for the inelastic bending deformation of such beams with account of the weakened resistance of their walls to the transverse shear. It is shown that, at discrete instants of time, the mechanical behavior of such structures can formally be described by the governing relations for composite beams made of nonlinear elastic anisotropic materials with a known initial stress state. The method of successive iterations, similar to the method of variable parameters of elasticity, is used to linearize the boundary-value problem at each instant of time. The bending deformation is investigated for homogeneous and reinforced cantilever and simply supported beams in creep under the action of a uniformly distributed transverse load. The cross sections of the beams considered are I-shaped. It is found that the use of the classical theory for such beams leads to the prediction of indefensibly underestimated flexibility, especially in long-term loading. It is shown that, in beams with reinforced load-carrying layers, the creep mainly develops due to the shear strains of walls. It is found that, in short- and long-term loadings of composite beams, the reinforcement structures rational by the criterion of minimum flexibility are different.

We consider a guaranteed control problem for a nonlinear distributed equation of diffusion type. The problem is essentially to construct a feedback control algorithm ensuring that the solution of a given equation tracks the solution of a similar equation subjected to an unknown disturbance. The case in which a discontinuous unbounded function can be a feasible disturbance is studied. We solve the problem under conditions of inaccurate measurement of solutions of each of the equations at discrete instants of time and indicate a solution algorithm robust under information noise and calculation errors.

The Harmonic Balance Method for nonlinear periodic flows is presented in this paper. Assuming a temporally periodic flow, a Fourier transformation is deployed in order to formulate a transient problem as a multiple quasi-steady-state problem. A solution of the obtained equations yields flow fields at discrete instants of time throughout a representative harmonic period, while still capturing the transient effect. The method is implemented in foam-extend, a community-driven fork of OpenFOAM\(^{\textregistered }\) and developed for multi-frequential use in turbomachinery applications. For validation, a 2D turbomachinery test case is used. Pump head, efficiency, and torque obtained with Harmonic Balance will be compared to a transient and steady-state simulation. Furthermore, pressure contours on rotor blades will be compared. And finally, in order to present the method’s efficiency along with its accuracy, a CPU time comparison will also be presented.

This paper presents a feedback control approach for the optimal power flow (OPF) problem which is suitable for operation in real-time. Traditionally, OPF is run in discrete instants of time in order to obtain the optimal operating points for the power system under given loading conditions at those instants. Between any two OPF runs, participation factors and/or other means are used to maintain generation-load balance. Hence, optimality is not maintained between OPF runs. This paper proposes the use of feedback control concepts to maintain optimality all the times while satisfying power system constraints. The feedback-control-based optimal power flow (FCOPF) is a reformulation of the well known gradient-based OPF in such a way that enables the use of feedback control concepts. Simulation results show that the proposed approach can keep track of the optimal operating point under varying loading conditions while meeting power system constraints.

For a discrete dynamical control system with noise, we consider the problem of retaining a phase point in a given family of sets at discrete instants of time. We analyze the case where the control vectogram is a polyhedron defined by a system of linear inequalities. We prove some properties of specific polyhedra satisfying the linearity condition, which allows us to obtain a condition of retention in the explicit form. Necessary and sufficient conditions for the possibility of retention are given. The results obtained are illustrated by an example.

Kalman Filtering Techniques