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Model Linearization
Published in Clarence W. de Silva, Modeling of Dynamic Systems with Engineering Applications, 2017
Limit cycles: A notable property of a nonlinear system is that its stability may depend on the system inputs and/or initial conditions. In particular, nonlinear devices may produce limit cycles. An example is given in Figure 4.1e on the phase plane (2D) of velocity versus displacement. A limit cycle is a closed trajectory in the state space that corresponds to sustained oscillations at a specific frequency and amplitude, without decay or growth. The amplitude of these oscillations is independent of the initial location from which the response started. In addition, an external input is not needed to sustain a limit cycle oscillation. In the case of a stable limit cycle, the response will move onto the limit cycle irrespective of the location in the neighborhood of the limit cycle from which the response was initiated (see Figure 4.1e). In the case of an unstable limit cycle, the response will move away from it with the slightest disturbance.
Oscillators
Published in Arturo Buscarino, Luigi Fortuna, Mattia Frasca, ®and Laboratory Experiments, 2017
Arturo Buscarino, Luigi Fortuna, Mattia Frasca
A limit cycle can be stable (if all neighboring trajectories approach it as time goes to infinity), unstable (if, instead, all neighboring trajectories approach it as time goes to negative infinity), or neither stable nor unstable. The limit cycle obtained in system (4.1) for μ > 0 is stable.
Handling actuator magnitude and rate saturation in uncertain over-actuated systems: a modified projection algorithm approach
Published in International Journal of Control, 2022
Seyed Shahabaldin Tohidi, Yildiray Yildiz
Actuator constraints such as magnitude and rate limits play a prominent role in advanced control systems. These limits induce nonlinear behaviour which may lead to performance degradation, occurrence of limit cycles, multiple equilibria, and even instability (Khalil, 2002; Tarbouriech et al., 2011). Actuator rate limits, specifically, introduce phase lags, which act as time delays, that can lead to persistent undesired oscillations called Pilot Induced Oscillations (PIO) (Acosta et al., 2015; Queinnec et al., 2017; Tohidi et al., 2018; Yildiz & Kolmanovsky, 2011a, 2011b, 2010; Yildiz et al., 2011). These oscillations generally occur due to an abnormal coupling between the pilot and the aircraft, instigated by various factors such as high pilot gains, actuator rate saturation and control mode switch (McRuer, 1995).
Robust limit cycle control in a class of nonlinear discrete-time systems
Published in International Journal of Systems Science, 2018
Ali Reza Hakimi, Tahereh Binazadeh
Limit cycle is an isolated periodic orbit in the phase plane which is an important phenomenon in nonlinear dynamical systems (Haddad & Chellaboina, 2008; Khalil & Grizzle, 2002). If the trajectories beginning near the limit cycle converge to it, then the limit cycle is attractive. This is a very rich dynamical behaviour which has the numerous applications in engineering systems like walking and running (Geyer, Seyfarth, & Blickhan, 2006; Laszlo, van de Panne, & Fiume, 1996; McGeer, 1990), power converters (Benmiloud & Benalia, 2016; Oviedo, Vazquez, & Femat, 2017), satellite altitude control (Clark, 1970), aero elastic problems (Bialy, Chakraborty, Cekic, & Dixon, 2016; Khalid & Akhtar, 2017), boiling-water reactors (Farawila & Pruitt, 2006), diffusively coupled models (Shafi, Arcak, Jovanović, & Packard, 2013), hybrid systems (Benmiloud, Benalia, Djemai, & Defoort, 2017; Flieller, Riedinger, & Louis, 2006) and port Hamiltonian systems (Aguilar-Ibañez, Mendoza-Mendoza, Martinez, Jesus Rubio, & Suarez-Castanon, 2015b).
Proprioceptive feedback design for gait synchronization in collective undulatory robots
Published in Advanced Robotics, 2022
Zhuonan Hao, Wei Zhou, Nick Gravish
The mathematical framework for designing and analyzing the synchronization of undulatory locomotion relies on the study of nonlinear oscillators. An oscillator is an autonomous dynamical system which exhibits a stable limit cycle attractor [14,15]. A stable limit cycle is a closed trajectory in the phase-space of the dynamical system in which a surrounding region of initial conditions is attracted onto the limit cycle. The closed trajectory of the limit cycle implies that the system's state variables will be periodic, i.e. the system in steady state will oscillate.