Abstract
Identifying new stable dynamical systems, such as generic stable mechanical or electrical control systems, requires questing for the desired systems parameters that introduce such systems. In this paper, a systematic approach to construct generic stable dynamical systems is proposed. In fact, our approach is based on a simple identification method in which we intervene directly with the dynamics of our system by considering a continuous \begin{document}$1$\end{document} -parameter family of system parameters, being parametrized by a positive real variable \begin{document}$\ell$\end{document} , and then identify the desired parameters that introduce a generic stable dynamical system by analyzing the solutions of a special system of nonlinear functional-differential equations associated with the \begin{document}$\ell$\end{document} -varying parameters. We have also investigated the reliability and capability of our proposed approach. To illustrate the utility of our result and as some applications of the nonlinear differential approach proposed in this paper, we conclude with considering a class of coupled spring-mass-dashpot systems, as well as the compartmental systems - the latter of which provide a mathematical model for many complex biological and physical processes having several distinct but interacting phases.
Highlights
The question as to whether a system of differential equations has stable solutions is of vital importance in engineering where it occurs in the investigation of mechanical, control, and electrical systems
Following our differential approach, we can identify with any desired accuracy the system parameters of a generic stable system; i.e., a system whose characteristic polynomial has N simple roots rk(λ) with negative real parts – from the practical point of view, it is important to note here that to the dynamics of such a system, ignoring the input, there corresponds a homogeneous differential equation of the same order N which has N independent solutions of the following forms: exp(rk(λ)t), if rk(λ) is real, or exp(Re(rk(λ))t) cos(Im(rk(λ))t) and exp(Re(rk(λ))t) sin(Im(rk(λ))t) otherwise; i.e., N independent solutions in the form of exponential decays with bounded functional multiples
We have proposed a nonlinear differential approach for identification of stable dynamical systems
Summary
The question as to whether a system of differential equations has stable solutions is of vital importance in engineering where it occurs in the investigation of mechanical, control, and electrical systems (for more details, see e.g. [23], [24], [27] and the references given therein). The important point to note here is that the constant coefficients αi’s appearing in the differential equation (1) are (given by) the parameters of the system whose dynamics is modeled by (1). We wish to investigate as to how one can systematically modify a system being modeled by (1) by varying (some or all of) the system parameters in order to construct a stable system This is precisely the situation that occurs in the designing of new stable mechanical or electrical control systems. To this end, following the original ideas of a recent work by Calogero [4] –instead of a test-bench approach as mentioned above, we take a nonlinear differential approach which is based on a simple identification method. Some other interesting topics for future work in this area are given in the conclusion (Sect. 5) of the paper
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