A constructive solution for stabilization via immersion and invariance: The cart and pendulum system

Acosta, José Ángel; Ortega, Roméo; Astolfi, Alessandro; Sarras, Ioannis

doi:10.1016/j.automatica.2008.01.006

Cited by 62 publications

(35 citation statements)

References 5 publications

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“…From the above, it follows that V T ( p) < V T ( p(0)) and, from its own definition, V T (t) is radially bounded 2 Notice that by definition…”

Section: Control Of the Cart Pole Systemmentioning

confidence: 91%

“…The first consists of the upward swing of the pendulum from the hanging position to the upright position. In general, this problem has been tackled by using methods based on energy control and hybrid schemes [2,3,8,9,22,26,31,44,45] . The second issue arises when the pendulum is located somewhere in the upper-half plane, and the goal is bringing it to its unstable equilibrium point.…”

Section: Introductionmentioning

confidence: 99%

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Stabilization of the cart pole system: by sliding mode control

Aguilar-Ibáñez¹,

Mendoza-Mendoza²,

Dávila³

2014

Nonlinear Dyn

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This paper presents a control strategy designed as a combination of a PD controller and a twistinglike algorithm to stabilize the damped cart pole system, provided that the pendulum is initially placed within the upper-half plane. To develop the strategy, the original system is transformed into a four-order chain of integrator form, where the damping force is included through an additional nonlinear perturbation. The strategy consists of simultaneously bringing the position and velocity of the pendulum to within a compact region by applying the PD controller. Meanwhile, the system state variables are brought to the origin by the twisting-like algorithm. The corresponding convergence analysis is done using several Lyapunov functions. The control strategy is illustrated with numerical simulations.

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“…From the above, it follows that V T ( p) < V T ( p(0)) and, from its own definition, V T (t) is radially bounded 2 Notice that by definition…”

Section: Control Of the Cart Pole Systemmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Stabilization of the cart pole system: by sliding mode control

Aguilar-Ibáñez¹,

Mendoza-Mendoza²,

Dávila³

2014

Nonlinear Dyn

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“…First, the application of a (collocated) partial feedback linearization stage, à la [22]. Second, following [1], the identification of conditions on the inertia matrix and the potential energy function that ensure the Lagrangian structure is preserved. As a corollary of the Lagrangian structure preservation two new passive outputs are easily identified.…”

Section: Introductionmentioning

confidence: 99%

Global Stabilisation of Underactuated Mechanical Systems via PID Passivity-Based Control

et al. 2017

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In this note we identify a class of underactuated mechanical systems whose desired constant equilibrium position can be globally stabilised with the ubiquitous PID controller. The class is characterised via some easily verifiable conditions on the systems inertia matrix and potential energy function, which are satisfied by many benchmark examples. The design proceeds in two main steps, first, the definition of two new passive outputs whose weighted sum defines the signal around which the PID is added. Second, the observation that it is possible to construct a Lyapunov function for the desired equilibrium via a suitable choice of the aforementioned weights and the PID gains and initial conditions. The results reported here follow the same research line as [7] and [20]-bridging the gap between the Hamiltonian and the Lagrangian formulations used, correspondingly, in these papers. Two additional improvements to our previous works are the removal of a non-robust cancellation of a potential energy term and the establishment of equilibrium attractivity under weaker assumptions.

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“…An eminent characteristic of physical pendulum equation has a rich content of nonlinear properties which are suitable for a detailed investigating various dynamical states. We therefore think it is worthwhile to undertake a detailed discussion of System (1). System (1) without vertically vibrated axis or vibration of angle γ has been extensively studied, for examples, as α = 0 and without the suspension axis vibrations, D'Humieres et al [10] and Landa [21] gave an experimental study of the chaotic states and the routes to chaos and showed the symmetry breaking as a precursor to the period-doubling route for chaos, intermittent behavior and period-triple bifurcations.…”

Section: Introductionmentioning

confidence: 99%

Complex dynamics in physical pendulum equation with suspension axis vibrations

Deng

Jing

2009

Acta Math. Appl. Sin. Engl. Ser.

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The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov's method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov's method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω + εν, n = 1 − 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 − 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f 0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from periodone orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven't find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.

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A constructive solution for stabilization via immersion and invariance: The cart and pendulum system

Cited by 62 publications

References 5 publications

Stabilization of the cart pole system: by sliding mode control

Stabilization of the cart pole system: by sliding mode control

Global Stabilisation of Underactuated Mechanical Systems via PID Passivity-Based Control

Complex dynamics in physical pendulum equation with suspension axis vibrations

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