Monodromy In Systems Research Articles

Fast-Slow Scale Bifurcation in Higher Order Open Loop Current-Mode Controlled DC-DC Converters

Higher order converters, without an external voltage control loop, have been known not to exhibit a fast-slow scale bifurcation interaction, as the slow-scale bifurcation has generally been believed to be caused by parameter changes in the outer loop. In this paper, it is shown that a current-mode controlled Ćuk converter can exhibit an interaction between fast-scale and slow-scale bifurcations even in the absence of this closed outer voltage control loop. The phenomenon is probed using an approach based on the system's monodromy matrix that does not only predict this instability but also provides a systematic method for the development of new control strategies to avoid the onset of this bifurcation. Analytical and numerical results prove that the new controller greatly extends the stable region of operation of the converter.

Fractional monodromy in systems with coupled angular momenta

We present a one-parameter family of systems with fractional monodromy, which arises from a 1:2 diagonal action of a dynamical symmetry SO(2). In a regime of adiabatic separation of slow and fast motions, we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi-quantum spectra.

Linear Time-Invariant Milling Models Applicable to Chatter Suppression System Design

Milling models, applicable to chatter suppression system design, are derived by transforming an established nonlinear milling model into linear constant coefficient discrete time models. The derivation initially utilizes either a modified Tustin or zero-order-hold approximation to create a discrete time approximation of the original model. The resulting periodically time varying system, which includes time delays, is then transformed into a system with constants coefficient and no time delays by adding delay states to the system and computing the system's monodromy matrix. The resulting linear models are then compared to the established nonlinear model. The linear models, while of high order, can be used directly to design chatter suppression systems with traditional linear control system design techniques. An example chatter control system is designed using an optimal regulator. The example controller reduces chatter in a milling system with a four tooth cutter in an up-milling configuration.

A reliable and efficient procedure for oscillator PPV computation, with phase noise macromodeling applications

The main effort in oscillator phase noise calculation and macromodeling lies in computing a vector function called the perturbation projection vector (PPV). Current techniques for PPV calculation use time-domain numerics to generate the system's monodromy matrix, followed by full or partial eigenanalysis. We present superior methods that find the PPV using only a single linear solution of the oscillator's time- or frequency-domain steady-state Jacobian matrix. The new methods are better suited for implementation in existing tools with harmonic balance or shooting capabilities (especially those incorporating fast variants), and can also be more accurate than explicit eigenanalysis. A key advantage is that they dispense with the need to select the correct one eigenfunction from amongst a potentially large set of choices, an issue that explicit eigencalculation-based methods have to face. We illustrate the new methods in detail using LC and ring oscillators.