Exact stability test of neutral delay differential equations via a rough estimation of the testing integral

Abstract

In this paper, we study the problem of stability test of neutral delay differential equations. Firstly we prove that the Mikhailov stability criterion and its equivalent integral form for ordinary differential equations hold for neutral delay differential equations in general form. The criteria are simple in form that is easy for numerical implementation. However, the criteria are characterized by an auxiliary function associated with the characteristic function, not by the characteristic function itself. To reduce the computational complexity, we further prove that a Mikhailov-type criterion in terms of the characteristic function holds. With this new criterion, the stability of a given neutral delay differential equation can be tested with a rough estimation of the testing integral. Thus, the computational complexity and computational cost can be greatly reduced. As two applications of these criteria, we firstly propose a numerical scheme for calculating the rightmost characteristic root(s) as well as the characteristic roots other than the rightmost roots of a given neutral delay differential equation, demonstrated with two examples. Then we derive a graphical stability criterion. With this graphical stability criterion, it is not required to know the exact curve of the Nyquist plot, but just to know whether the Nyquist plot encircles the origin of the complex plane or not.

Suppression of oscillatory congestion via trunk link bandwidth and control gain in star network

“…11 that the strongest "damping" is achieved at C = 4.41 for any C provided that the equilibrium is stable. Figures 11 and 12 are plotted using DDE-BIFTOOL [34] and the integral evaluation method [35][36][37] , respectively. First, we calculate the largest real part of eigenvalue for a specific point (C, τ ) = (3, 0.5) as −1.113 501 6 based on the method in Ref.…”

“…First, we calculate the largest real part of eigenvalue for a specific point (C, τ ) = (3, 0.5) as −1.113 501 6 based on the method in Ref. [36]. The characteristic equation is given by f (λ) = λ 2 − 2a 1 λe −λτ + 2(a 2 1 − a 2 2 )e −2λτ , where a 1 and a 2 are shown in…”

Balancing a wheeled inverted pendulum with a single accelerometer in the presence of time delay

“…Figure 5 shows the stability charts obtained by the stability test method for NDDE (Xu and Wang, 2014), in the plane of the parameters P and τ , for different values of L . The results show that the WIP can be stabilized by adjusting the values of P and τ , and that the location of the accelerometer is relevant; it can significantly increase the stable region, while placing the accelerometer at the wheel’s center can not stabilize the WIP.…”

“…We will also discuss how the time delay in the control loop affects the choices. Figure 5 shows the stability charts obtained by the stability test method for NDDE (Xu and Wang, 2014), in the plane of the parameters P and , for different values of L. The results show that the WIP can be stabilized by adjusting the values of P and , and that the location of the accelerometer is relevant; it can significantly increase the stable region, while placing the accelerometer at the wheel's center can not stabilize the WIP. Figure 6 presents the time history of the linearized WIP for L ¼ 0.8 m, P ¼ 0.05 N, ¼ 0.005 s, where by choosing a time delay within the stability range, the WIP is balanced successfully and the convergence is fast.…”

Root Loci Analysis of a Simple Quasi-Polynomial And Its Use For Relay-Based Parameter Identification of a Delayed Infinite-Dimensional Heat-Exchanger Process

“…Recently, several ad-hoc results for single 17 and multiple real 20,23 prescribed poles or even single a complex conjugate pair 24 guarantying their dominancy have been derived; however, they usually levy large computational burden. Alternatively, the root dominance can be a posteriori checked using the argument principle (i.e., the Mikhailov curve based) approach 25 or via the solution of a special convolution integral 26 , which requires an advanced mathematical effort as well. Whenever the direct assignment is not satisfactory, a numerical spectrum optimization can be made, e.g., by the quasi-continuous shifting of the roots 27,28 or using its combination with the minimization of a specific fitness function reflecting the remaining spectrum, robustness issues, etc.…”