Bifurcation Analysis and Dynamic Behaviour of an Inverted Pendulum with Bounded Control

Николов, Светослав; Nedev, Valentin

doi:10.1515/jtam-2016-0002

Cited by 6 publications

(3 citation statements)

References 19 publications

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“…A similar work was realized on the Furuta pendulum by analyzing the Hopf bifurcation [59]. Recently, Nikolov and Nedev [60] analyzed bifurcation and dynamic behavior of the IWP with bounded control by means of the theory of Poincaré-Andronov-Hopf.…”

Section: Literature Reviewmentioning

confidence: 92%

Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC

et al. 2017

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This paper deals with the tracking approach of a self-generated stable limit cycle for an underactuated mechanical system: The Inertia Wheel Inverted Pendulum (IWIP). Such system is subject to unilateral constraints limiting its swing motion. It is known that an Interconnection and Damping Assignment-Passivity Based Control (IDA-PBC) can be employed to control such pendulum to its upright position. In this work, we briefly show first that the IWIP can generate a stable period-1 limit cycle through a Hopf bifurcation by varying some gain parameter of the IDA-PBC. Thus, such self-generated limit cycle is used as a reference trajectory, which is chosen to be tracked by the IWIP. To achieve the tracking problem, a supplementary control input is added. Such tracking problem is reformulated as an asymptotic stabilization of the tracking error. Our fundamental approach hinges mainly on the use of the S-procedure to introduce the unilateral constraints, and the Schur complement and the matrix inversion lemma to transform Bilinear Matrix Inequalities (BMI) into Linear Matrix Inequalities (LMI). Several simulations have been presented to corroborate the mathematical results and to show the efficiency of the proposed tracking scheme of the self-generated stable limit cycle of the controlled IWIP, even if it is subject to external disturbances, or in the presence of uncertainties in the friction parameters.

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Section: Literature Reviewmentioning

confidence: 92%

Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC

et al. 2017

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“…It is seen that for first and second fixed points we have: r (1) = a + b, R (1) = −a; r (2) = −a − b; R (2) = a + 2b. According to [12,23,40,41], we obtain the formula describing the first Lyapunov value for our system and calculate its value in the calculated bifurcation points (for more details, see Appendix A). Thus, for L 1 , we write:…”

Section: Analytical Investigation 21 Local Analysis and First Lyapumentioning

confidence: 99%

Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

Николов

Vassilev

2021

Mathematics

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In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs-Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain a new analytical formula for the first Lyapunov’s (focal) value at the boundary of stability of the corresponding equilibrium state. On the other hand, the global analysis reveals that the system may exhibit the phenomena of Shilnikov chaos. Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator. Finally, it is found that for some new combinations of parameters, the system demonstrates chaotic behavior and transition from chaos to regular behavior is realized through inverse period-doubling bifurcations.

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“…A similar work was realized on the Furuta pendulum by analyzing the 25 Hopf bifurcation [Pagano et al, 2000]. Recently, Nikolov and Nedev [Nikolov & Nedev, 2016] analyzed bifurcation and dynamic behavior of the IWIP with bounded control by means of the theory of Poincaré-Andronov-Hopf (PAH). In this study we investigated the stabilization of oscillations by a passivity-based design control, namely Interconnection and Damping assignment Passivity Based Control (IDA-PBC) introduced first in [Ortega 30 et al, 2002] and revisited in [Khraief-Haddad et al, 2014] and [Khraief-Haddad et al, 2015].…”

mentioning

confidence: 95%

From Hopf Bifurcation to Limit Cycles Control in Underactuated Mechanical Systems

Haddad

Belghith

Gritli

et al. 2017

Int. J. Bifurcation Chaos

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This paper deals with the problem of obtaining stable and robust oscillations of underactuated mechanical systems. It is concerned with the Hopf bifurcation analysis of a Controlled Inertia Wheel Inverted Pendulum (C-IWIP). Firstly, the stabilization was achieved with a control law based on the Interconnection, Damping, Assignment Passive Based Control method (IDA-PBC). Interestingly, the considered closed loop system exhibits both supercritical and subcritical Hopf bifurcation for certain gains of the control law. Secondly, we used the center manifold theorem and the normal form technique to study the stability and instability of limit cycles emerging from the Hopf bifurcation. Finally, numerical simulations were conducted to validate the analytical results in order to prove that with IDA-PBC we can control not only the unstable equilibrium but also some trajectories such us limit cycles.

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Bifurcation Analysis and Dynamic Behaviour of an Inverted Pendulum with Bounded Control

Cited by 6 publications

References 19 publications

Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC

Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC

Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

From Hopf Bifurcation to Limit Cycles Control in Underactuated Mechanical Systems

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