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Ordinary differential equations
Published in Victor A. Bloomfield, Using R for Numerical Analysis in Science and Engineering, 2018
Delay differential equations (DDEs) differ from ODEs in that they contain derivatives that depend on the values of the variables at previous times. The initial conditions for DDEs must specify not only the values at time = 0, but also at times < 0 back to the time of the longest lag in the problem. DDEs also present difficulties in that they often have discontinuities in low-order derivatives; e.g., a constant value up to t = 0, followed by a non-zero slope for t > 0. Therefore, numerical methods developed for ODEs must be modified to deal with DDEs. A useful discussion of DDEs and their solutions in the context of MATLAB® is Shampine and Thompson 2000 (http://www.radford.edu/thompson/webddes/tutorial.html).
A hybrid approach to parameter identification of linear delay differential equations involving multiple delays
Published in International Journal of Control, 2018
This paper deals with the parameter estimation of linear multi-delay systems. Delay differential equations (DDEs) arise in various fields of applications including mathematical biology, industrial processes, transmission lines, manufacturing systems, communication and information technologies (Smith, 2010). As a consequence, stability analysis, optimal control, and parameter identification of this class of systems are of theoretical and practical importance. It is worth noting that the parameter identification of different kinds of time-delay systems is an essential issue in control system analysis and design. Although the parameter estimation of DDEs is one of the most attractive tasks in the context of control theory, a few research works have been committed to the investigation and numerical treatment of the mentioned systems (Ardekani, Samavat, & Rahmani, 1991; Hwang & Chen, 1985; Horng & Chou,1985; Horng & Chou, 1986; Wang, Jan, & Chang, 1987; Yang & Chen, 1987). In general, they are difficult to be analysed or identified. It should be pointed out that all of the computational procedures presented in the mentioned works do not provide accurate results to DDEs. This issue is essentially due to the fact that the output function associated with the input function is a piecewise smooth function. It is important to point out that due to the lack of smoothness of the output function, smooth basis functions such as Legendre's polynomials and Chebyshev's polynomials are not inherently able to properly model the discontinuities points that arise in either state trajectory or control input. Recently, Belkoura and Orlov (2002), Belkoura, Richard, and Fliess (2009), and Belkoura (2012) have made excellent progress in this respect. Identifiability and algebraic identification of time-delay systems have been studied in Belkoura (2012). Identifiability results are first given for linear delay systems expressed by convolution equations. Online algorithms are next developed for both parameters and delay estimation. Sufficient conditions for the identifiability of a general class of systems described by convolution equations have been proposed. Besides, an algebraic approach for the identification of time-delay systems based on both structured inputs and arbitrary input–output trajectories has been developed. The simulation results with noisy data demonstrate the effectiveness of the suggested method.
A class of nonlinear optimal control problems governed by Fredholm integro-differential equations with delay
Published in International Journal of Control, 2020
Hamid Reza Marzban, Mehrdad Rostami Ashani
In the mathematical explanation of a physical process, it is generally assumed that the dynamical behaviour of a phenomena depends exclusively on the present state and is independent of its past. This assumption is verified for an extensive class of dynamical systems. However, there are many real-life phenomena and practical systems where this assumption is not valid. Accordingly, the use of a classical model to analyse the system under investigation may lead to poor performance and undesired behaviour such as oscillation and instability. Delay differential equations (DDEs) provide a flexible framework to represent the dynamical behaviour of a large class of processes. Time delays occur so often in almost every situation, that to ignore them is to ignore reality (Kuang, 1993). The theory of DDEs is a very extensive field which has numerous applications in various fields of science and engineering, such as mathematical biology, population dynamics, traffic flows, chemical processes, pneumatic and hydraulic systems, electrodynamics, quantum mechanics, robotics, and communication networks (Smith, 2010). Time delays have a significant influence on the stability and controllability of the system under study. As a consequence, the presence and effects of time delays cannot be ignored. It is known that the existence of delay makes the method of solution much more complicated. As a concrete example of time-delay systems, consider the structure of supply chains described below. Supply chains are an interconnection of different dynamics contributed by customers, suppliers, manufacturing units, assembly lines, parallel running processes, and sources (Sipahi & Delice, 2010). It should be pointed out that one of the most significant goals in managing supply chains is to adjust the inventory levels while successfully responding to customer requirements. It may appear like a simple problem, but in reality, supply chain management is a challenging problem in presence of delays. The existence of delays may cause poor performance and unexpected behaviour such as inventory oscillations. Delays occur in supply chains owing to decision-making, production lead-time, transportation times, and lags in flow of information. As a result, optimal control of this class of systems is of theoretical and practical importance. It is well-known that except for some simple cases, it is either highly difficult or actually impractical to analytically solve a nonlinear delay differential equation. The situation becomes much more complicated when we are concerned with a nonlinear optimal control problem governed by integro-differential equation containing delay.