Recently, a novel discrete-time nonlinear limit cycle model predictive controller for harmonic compensation has been proposed. Its compensating action is achieved by using the dynamics of a ...supercritical Neimark-Sacker bifurcation normal form at the core of its cost function. This work aims to extend this approach's applicability by analyzing its stability. This is accomplished by identifying the normal form's region of attraction and final set, which enables the use of LaSalle's invariance principle. These results are then extended to the proposed controller under ideal conditions, i.e., zero-cost solutions with predictable disturbances. For nonideal scenarios, i.e., solutions with unpredictable disturbances and cost restrictions, conditions are developed to ensure that the closed-loop system remains inside the normal form's region of attraction. These findings are tested under nonideal conditions in a power systems application example. The results show successful power quality compensation and a satisfactory resilient behavior of the closed loop within the margins developed during this work.
In this paper, we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincaré maps. In particular, we ...prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding zero-input fixed point of the associated forced Poincaré map. This result extends the classical Poincaré analysis for asymptotic stability of periodic solutions to establish orbital ISS of such solutions under external excitation. In our proof, we define the forced Poincaré map, and use it to construct ISS estimates for the periodic orbit in terms of ISS estimates of this map under mild assumptions on the input signals. As a consequence of the availability of these estimates, the equivalence between exponential stability (ES) of the fixed point of the zero-input (unforced) Poincaré map and the ES of the corresponding orbit is recovered. The results can be applied naturally to study the robustness of periodic orbits of continuous-time systems as well. Although our motivation for extending classical Poincaré analysis to address ISS stems from the need to design robust controllers for limit-cycle walking and running robots, the results are applicable to a much broader class of systems that exhibit periodic solutions.
The digital low dropout regulator (D-LDO) has drawn significant attention recently for its low-voltage operation and process-scalability. However, the D-LDO inherently suffers from limit cycle ...oscillation (LCO). To address this issue, the modes and amplitudes of LCO are calculated in this work and verified by SPICE simulation in a 65-nm CMOS process. An LCO reduction technique for the D-LDO is then proposed, by adding two unit power transistors in parallel with the main power MOS array as a feedforward path. This technique sets the LCO mode to 1 and effectively reduces the ripple amplitude for a wide (0.5-20 mA) load current range. When compared with the dead-zone scheme, this technique minimizes LCO with negligible circuit complexity and design difficulty.
When designing feedback controllers to achieve periodic movements, a reference trajectory generator for oscillations is an important component. Using autonomous oscillators to this effect, rather ...than directly crafting periodic signals, may allow for systematic coordination in a distributed manner and storage of multiple motion patterns within the nonlinear dynamics, with potential extensions to adaptive mode switching through sensory feedback. This article proposes a method for designing a distributed network that possesses multiple stable limit cycles from which various output patterns are generated with prescribed frequency, amplitude, temporal shapes, and phase coordination. In particular, we adopt, as the basic dynamical unit, a simple nonlinear oscillator with a scalar complex state variable, and derive conditions for their distributed interconnections to result in a network that embeds desired periodic solutions with orbital stability. We show that the frequencies and phases of target oscillations are encoded into the network connectivity matrix as its eigenvalues and eigenvectors, respectively. Various design examples will illustrate the proposed method, including generation of human gaits for walking and running.
In this paper, we study the number of limit cycles that can bifurcate from a periodic annulus in discontinuous planar piecewise linear Hamiltonian differential system with three zones separated by ...two parallel straight lines, such that the linear differential systems that define the piecewise one have a center and two saddles. That is, the linear differential system in the region between the two parallel lines (called of central subsystem) has a center and the others subsystems have saddles. We prove that if the central subsystem has a real or a boundary center, then at least six limit cycles can bifurcate from the periodic annulus by linear perturbations. Four passing through the three zones and two passing through two zones. Now, if the central subsystem has a virtual center, then at leas four limit cycles can bifurcate from the periodic annulus by linear perturbations, three passing through the three zones and one passing through two zones. For this, we obtain a normal form for these piecewise Hamiltonian systems and study the number of zeros of its Melnikov functions defined in two and three zones.
This article reveals, analyzes, and proposes the method to mitigate nonlinear effects of multisampled multiupdate (MSMU) digital pulsewidth modulation (PS-DPWM) that appear in unbalanced multicell ...voltage-source converters (MC-VSCs). For balanced MC-VSCs, the harmonic cancellation of PS-DPWM allows for an increase in the sampling frequency, ensuring that the average current is acquired at the peaks, valleys, and intersections of all the triangular carriers. For unbalanced operation, which is typically encountered in practice, e.g., due to cell voltage mismatch in multilevel MC-VSCs and inductance mismatch in interleaved MC-VSCs, harmonic cancellation of PS-DPWM is compromised and, thus, the increased sampling frequency brings switching ripple in the feedback signal. Since in MSMU control the modulating signal is also updated at peaks, valleys, and intersections of all the carriers, this may cause vertical intersections between the modulating signal and the carriers, resulting in specific nonlinear effects. The nonlinearities are shown to introduce limit-cycle oscillations (LCOs) and output waveform distortion. A method to prevent such detrimental impact of MSMU-PS-PWM is also proposed. A simple analytical procedure is proposed to quantify the analyzed nonlinear effects, revealing that they are more emphasized for higher levels of imbalance and control bandwidth. Moreover, the modulator nonlinearity is shown to decrease as the number of cells increases. The analyses are verified in simulations and experiments, using laboratory prototypes of three- and four-level MC-VSCs.
Nonlinear stability becomes critical when periodic and aperiodic oscillations arise and are important for the safe operation of nuclear reactors. The linear stability analysis and Hopf bifurcation ...are well studied in the context of nuclear reactors and fail to detect higher-order nonlinear entities. A reduced-order model which couples neutron dynamics with thermal-hydraulics is used in this work. Hopf bifurcation and limit cycles were reported earlier; however, the bifurcation analysis of limit cycles with two free parameters was not provided in the past and is investigated here. These co-dimension two bifurcations of limit cycles, which define the origin of several bifurcations such as (limit point (LPC), period doubling (PD), and Neimark-Sacker (NS) bifurcation of limit cycles), are our focus. We analyze the bifurcation starting from the Generalized Hopf (GH) and observe R1 resonance bifurcations along with the cusp bifurcation of limit cycles (CPC). We reveal dynamics near the R1 and quasiperiodic behavior are present in the vicinity. We observe cascading of R1 bifurcation due to multiple LPCs occurring in the supercritical Hopf region. We also detect CPC, which changes the direction of the LPC curve, which we call the global stability boundary. We present aperiodic and uncertain oscillations near R1, and for a safer operation, we should understand the existence of higher-order bifurcations.
In this article, we study the formation problem for a group of mobile agents in a plane, in which the agents are required to maintain a distribution pattern, as well as to rotate around or remain ...static relative to a static/moving target. The prescribed distribution pattern is a class of general formations where the distances between neighboring agents or the distances between each agent and the target do not need to be equal. Each agent is modeled as a double integrator and can merely perceive the relative information of the target and its neighbors, and the acceleration of the target. In order to solve the formation problem, a limit-cycle-based controller design is delivered. We divide the overall control objective into two subobjectives, where the first is target circling that each agent keeps its own desired distance to the static/moving target as well as rotating around or remaining static relative to the target as expected, and the second is distribution adjustment that each agent maintains the desired distance to its neighbors. Then, we propose a controller comprised of two parts, where a limit cycle oscillator named a converging part is designed to deal with the first subobjective, while a layout part is introduced to address the second subobjective. One key merit of the controller is that it can be implemented by each agent in its local frame so that only local information is utilized without knowing global information. Theoretical analysis of the convergence to the desired formation, of which the agents are required to be evenly distributed on a circle around the target, is provided for the multiagent system under the proposed controller. Numerical simulations are given to validate the effectiveness of the proposed controller for the cases of general formations, and to show that no collision between agents ever takes place throughout the system's evolution.
Climate change poses challenges in classifying ecosystem dynamics, as they are influenced by shifting dynamics resulting from changes in climate forces and meteorological variables, including ...temperature and water availability. To address this, our study presents a novel approach using Continuous Wavelet Transform (CWT) and power spectrum analysis to classify vegetation dynamics, considering the time-dependent variability of ecosystem frequencies. We applied our method to centred and standardized MODIS NDVI time series for the period 2000–2021, using an experimental field station in northern Patagonia as a case study. By performing a continuous wavelet transform on the data for each pixel, we obtained instantaneous power spectra, capturing variability across different dates and pixels. These spectrums were then consolidated into a comprehensive database, and subsequently classified using archetypal analysis. We identified a convex combination of archetypal spectrums that best represented the entire power spectrum database. Mapping the resulting archetypes and their weights in both space and time allowed us to explore pixels' variations in archetype weights in relation to factors such as time, topography, and climate. In addition, to examine the potential relationship between the NDVI time series and climate drivers, we computed the Average Cross-Wavelet Power Spectrum (ACWPS) to different climatic indices. Three archetypes were sufficient to explain the majority of power spectrum variability in the study area. These archetypes exhibited distinctive characteristics: 1) medium-frequency variability (2–4 years), 2) low-frequency variability (>4 years), and 3) an annual (i.e. seasonal) cycle with low-frequency variability. Spatially, the first two archetypes were predominantly observed in highland steppes, while the third archetype prevailed in lowland areas associated with meadows. At the beginning of the studied period, Archetypes 1 and 3 dominated, but after the Puyehue-Cordón Caulle Volcanic Complex eruption in 2011 their prominence diminished, and Archetype 2 became more prevalent in the whole study area. Finally, all three NDVI series representative of archetypes showed a relative peak at approximately four years, which could be linked to the Indian Ocean Dipole variability. These results highlight an abrupt shift in the system's behaviour, primarily related to changes in variability distribution rather than mean values. This disturbance-induced transition aligns with the theory of state and transitions in ecological system dynamics. We propose that the states in this model are not fixed but represent alternative dynamic behaviours, akin to different types of limit cycles. Consequently, employing a wavelet analysis-based classification method provides a robust means of studying and understanding such variability and transitions, thereby offering clarity and comprehension of ecosystem states. Notably, this methodology proves particularly effective for large databases of detailed time series.
•Classifies based on time and space location.•Divides typologies based on time-frequency.•Describes population dynamics transitions and changes.•Identifies changes in limit cycles of vegetation productivity.•Independent of mean and variance in vegetation index time series.
The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this paper, we study the interplay between the functional response of Holling type IV and ...both strong and weak Allee effects. The model investigated here presents complex dynamics and high codimension bifurcations. In particular, nilpotent cusp singularity of order 3 and degenerate Hopf bifurcation of codimension 3 are completely analyzed. Remarkably it is the first time that three limit cycles are discovered in predator-prey models with multiplicative Allee effects. Moreover, a new unfolding of nilpotent saddle of codimension 3 with a fixed invariant line is discovered and fully developed, and the existence of codimension 2 heteroclinic bifurcation is proven. Our work extends the existing results of predator-prey systems with Allee effects. The bifurcation analysis and diagram allow us to give biological interpretations of predator-prey interactions.
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