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Delay Differential Equations
Published in Federico Milano, Ioannis Dassios, Muyang Liu, Georgios Tzounas, Eigenvalue Problems in Power Systems, 2020
Federico Milano, Ioannis Dassios, Muyang Liu, Georgios Tzounas
The quasi-polynomial mapping-based root-finder (QPMR) is a technique developed by Vyhlídal and Zítek [173] and is a general tool for all kinds of NEPs. This method aims at computing all zeros for the following quasi-polynomial: h(s)=det(Δ(s)).
Optimal Simultaneous Multisurface and Multiobject Image Segmentation
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
Xiaodong Wu, Mona K. Garvin, Milan Sonka
The most related problem is the metric labeling problem in computer science or called the Markov Random Field (MRF) optimization problem in computer vision. Our EOSD problem is a substantial generalization of the metric labeling problem, in which λ = 1 and no region term is involved. The metric labeling problem captures a broad range of classification problems where the quality of a labeling depends on the pairwise relations between the underlying set of objects, such as image restoration [18, 19], image segmentation [20, 21, 22, 23], visual correspondence [24, 25], and deformable registration [26]. After being introduced by Kleinberg and Tardos [27], it has been studied extensively in theoretical computer science [28, 29, 30, 31]. The best-known approximation algorithm for the problem is an O(log L) (L is the number of labels, or in our case, L = Z) [30, 27] and has no Ω(logL)) approximation unless NP has quasi-polynomial time algorithms [29]. Due to the application nature of the problem, researchers in image processing and computer vision have also developed a variety of good heuristics that use classical combinatorial optimization techniques, such as network flow and local search (e.g., [18, 32, 21, 20, 24, 33]), for solving some special cases of the metric labeling problem.
New Methodology for Chatter Stability Analysis in Simultaneous Machining
Published in Bogdan M. Wilamowski, J. David Irwin, Control and Mechatronics, 2018
This is a three-dimensional optimization problem, in which τ1, τ2, and a are independent parameters, and |ℜe(sdom)| is a measure of chatter rejection, as explained earlier. CTCR paradigm helps us obtain the stability boundaries of this system in the time-delay domain, but obtaining the real part of the rightmost characteristic roots requires that we look inside each stability region. Finding the real part of rightmost characteristic root of a quasi-polynomial is still an open problem in mathematics. One of the few numerical schemes available in the literature to “approximate” the rightmost (dominant) roots for different values of time delays is given in [42] that will be used for this optimization problem. It is clear that the variation of τ1, τ2 as well as axial d-o-c, a, is continuous. Figure 7.12 displays an example case study of four-flute milling cutter, taken from [37]. For that figure, a grid-point evaluation of the objective function is performed, which results in the respective chatter rejection abilities. One can seek the most desirable points in the stable operating regions. At any point (τ1, τ2), we can characterize the tool geometry (i.e., the pitch angles θ1 and θ2) as well as the spindle speed N = 30/(τ1 + τ2).
Dynamic similarity approach to control system design: delayed PID control loop
Published in International Journal of Control, 2019
Pavel Zítek, Jaromír Fišer, Tomáš Vyhlídal
With the use of (7) or (8) together with (9), the control loop is described in a thoroughly dimensionless form. From various options of control loop operation, the disturbance rejection ability is evaluated by means of the IAE criterion, i.e. by the control error integral where is raised by a disturbance step change. By means of ρP, ρI, ρD tuning as low as possible, IAE is to be achieved. As Equations (7) –(10), the following transfer function is also introduced as dimensionless using operator corresponding to the frequency number Π6: . After closing the feedback, the control loop transfer function for disturbance d(t) and for the plant (7) then results as where the delay exponential is cancelled since . In a similar manner, the disturbance transfer function for the case with plant (8) is obtained. The denominators of these functions determine their characteristic quasi-polynomials. For the case of (12), the characteristic quasi-polynomial is as follows:
Genetic algorithm optimization to model business investment in fashion design
Published in International Journal of Management Science and Engineering Management, 2023
Khaled A. Al Utaibi, Robia Arif, Sadiq M. Sait, Ayesha Sohail, Mahnoor Awan
This equation is known as characteristic equation of equilibrium point. If exponential term is zero then it is the equation for find the eigenvalues of ordinary differential equation. The exponential term is called quasi polynomial. If the solution of Equation (13) is real and negative then the system at this equilibrium point is stable. If it gives positive real-value then system is unstable but if the value is zero then it is neither stable nor unstable for ordinary differential equation. In case of ordinary differential equation the order and number of roots are same but in delay differential equation quasi polynomials infinitely roots which may be in complex plane. So we use different conditions to prove the stability.
Stabilising PID controller for time-delay systems with guaranteed gain and phase margins
Published in International Journal of Systems Science, 2022
Figure 6 shows the real and imaginary parts of . It can be seen that the roots interlace. From Theorem 3.6, for , we have . Next, it is necessary to ensure that all roots of real and imaginary parts of are real. From (22), we get Applying Lemma 3.5 results in Changing the variable, , yields For this quasi-polynomial in , one has and . As well, the real and imaginary parts of are as follows: Now, we focus on the real part of . First, we find the roots of that are . Second, we choose because . For , according to (27), we have and . It can be seen that there are exactly seven roots in this interval. Thus, all roots of are real.