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Controller Tuning
Published in Raghunathan Rengaswamy, Babji Srinivasan, Nirav Pravinbhai Bhatt, Process Control Fundamentals, 2020
Raghunathan Rengaswamy, Babji Srinivasan, Nirav Pravinbhai Bhatt
The last aspect of time-delay systems is the stability of such systems. Because time-delay element in the denominator results in an infinite order polynomial, stability analysis is not amenable to the partial fraction expansion approach. The conceptualization discussed in Figure 6.7 (while useful for controller realization) is not useful because the stability of U(s) will depend on Ud(s), which cannot be decided a priori. At this point, we will introduce a new concept to study the stability of time-delay systems. This concept is the Nyquist stability criterion. It is derived using two results in complex analysis, the residue theorem and principle of arguments. Without going into all of the mathematical details, important ideas will now be presented. While the results are applicable for time-delay systems, the main concepts will be illustrated using the standard numerator and denominator polynomials. Consider a TF: F(s)=(s−z1)…(s−zm)(s−p1)…(s−pn)
Complex Analysis, Differential Equations, and Laplace Transformation
Published in Ramin S. Esfandiari, Bei Lu, Modeling and Analysis of Dynamic Systems, 2018
This chapter presents a review of complex analysis, differential equations, and Laplace transformation, providing the necessary background for a better understanding of various ideas and implementation of methods involved in the analysis of dynamic systems. Complex analysis comprises the study of complex numbers, complex variables, and complex functions. Ordinary differential equations (ODEs) arise in situations where the rate of change of a function with respect to its independent variable is involved. Differential equations are generally very difficult to solve, even for the simplest case of constant coefficients. To that end, Laplace transformation is used to solve initial-value problems (IVPs)—ODEs subjected to initial conditions—by transforming the data from time domain to the s-domain, where equations are algebraic and hence easier to work with. Transformation of the information from the s-domain back to time domain ultimately describes the solution of the IVP.
Complex Analysis
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
As we know the real analysis or calculus is the study of functions of real variables (f : R → R or f : R2 → R). Similarly complex analysis is the study of functions of complex variables f : C → C, where C denotes the set of complex numbers z = x + iy, x ∈ R, y ∈ R) and the imaginary number i=−1 is the root of the algebraic equation x2 + 1 = 0. It is well known that complex analysis was developed as a result of mathematical curiosity but subsequently it was found very useful in signal and image processing, fluid flow, quantum mechanics and many other areas of engineering.
Applications of q-derivative operator to subclasses of bi-univalent functions involving Gegenbauer polynomials
Published in Applied Mathematics in Science and Engineering, 2022
Qiuxia Hu, Timilehin Gideon Shaba, Jihad Younis, Bilal Khan, Wali Khan Mashwani, Murat Çağlar
Geometric Functions Theory is a fascinating area of research in Complex Analysis, with applications in a variety of mathematical areas, including Mathematical Physics. Researchers in the field of Complex Analysis have been looking into holomorphic functions because of their various applications in analytical solutions to problems like electrostatics and fluid mechanics.