Double Imaginary Root Degeneracies in Time-Delayed Systems and CTCR Treatment

The dynamics we treat here is a very special and degenerate class of linear time-invariant time-delayed systems (LTI-TDS) with commensurate delays, which exhibit a double imaginary root for a particular value of the delay. The stability behavior of the system within the immediate proximity of this parametric setting which creates the degenerate dynamics is investigated. Several recent investigations also handled this class of systems from the perspective of calculus of variations. We approach the same problem from a different angle, using a recent paradigm called Cluster Treatment of Characteristic Roots (CTCR). We convert one of the parameters in the system into a variable and perturb it around the degenerate point of interest, while simultaneously varying the delay. Clearly, only a particular selection of this arbitrary parameter and the delay enforce the degeneracy. All other adjacent points would be free of the mentioned degeneracy, and therefore can be handled with the CTCR paradigm. Analysis then reveals that the parametrically limiting stability behavior of the dynamics can be extracted by simply using CTCR. The results are shown to be very much aligned with the other investigations on the problem. Simplicity and numerical speed of CTCR may be considered as practical advantages in analyzing such systems. This approach also exhibits the capabilities of CTCR in handling these degenerate cases contrary to the convictions in earlier reports. An example case study is provided to demonstrate these features.

Abstract simplicity of locally compact Kac–Moody groups

In this paper, we establish that complete Kac–Moody groups over finite fields are abstractly simple. The proof makes essential use of Mathieu and Rousseau’s construction of complete Kac–Moody groups over fields. This construction has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.

Computation of Stability Delay Margin of Time-Delayed Generator Excitation Control System with a Stabilizing Transformer

This paper investigates the effect of time delays on the stability of a generator excitation control system compensated with a stabilizing transformer known as rate feedback stabilizer to damp out oscillations. The time delays are due to the use of measurement devices and communication links for data transfer. An analytical method is presented to compute the delay margin for stability. The delay margin is the maximum amount of time delay that the system can tolerate before it becomes unstable. First, without using any approximation, the transcendental characteristic equation is converted into a polynomial without the transcendentality such that its real roots coincide with the imaginary roots of the characteristic equation exactly. The resulting polynomial also enables us to easily determine the delay dependency of the system stability and the sensitivities of crossing roots with respect to the time delay. Then, an expression in terms of system parameters and imaginary root of the characteristic equation is derived for computing the delay margin. Theoretical delay margins are computed for a wide range of controller gains and their accuracy is verified by performing simulation studies. Results indicate that the addition of a stabilizing transformer to the excitation system increases the delay margin and improves the system damping significantly.

Computer Aided Geometric Method of Forward Kinematics Analysis of Multi-Legged Walking Robots

The high order equation causing by analytic method in multi-legged robot forward kinematics analysis may have imaginary root, repeated root, extraneous root or even lost solution. A system based on the theory of computer aided geometric method is proposed. Consideration with the internal structural constraint relations of multi-legged walking robots, the solidworks model was constructed and Visual Basic develop platform was adopted to fulfill the secondary development of solidworks. A system of forward kinematics analysis of multi-legged walking robots is established. The example validates that the system is simple and effective for all reptiles-like quadruped walking robot.

On the characters of affine Kac–Moody groups

Let G be an affine Kac–Moody group over ℂ, and V∞ an integrable simple quotient of a Verma module for g. Let Gmin be the subgroup of G generated by the maximal algebraic torus T, and the real root subgroups.It is shown that (the least positive imaginary root) gives a character δ∈Hom(G, ℂ*) such that the pointwise character χ∞ of V∞ may be defined on Gmin ∩ G>1.