Controlling an Inverted Pendulum with Bounded Controls

Alonso, Diego; Paolini, Eduardo E.; Moiola, Jorge L.

doi:10.1007/3-540-45606-6_1

Cited by 14 publications

(18 citation statements)

References 15 publications

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“…where ϕ 1 is the angle of the arm (ϕ 1 = 0 at the upright position), ϕ 2 is the angular velocity of the arm, ϕ 3 is the angular velocity of the disk with respect to arm, u is the control input (voltage applied to the motor), and q 1 , q 2 , q 3 and ρ > 0 are constant coefficients, derived from physical parameters [21]. Here, we note that the disk position is not considered as a stable variable, because it is irrelevant for the stabilization of the pendulum in the inverted position.…”

Section: Problem Formulationmentioning

confidence: 99%

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

Николов¹,

Nedev²

2016

Journal of Theoretical and Applied Mechanics

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Abstract. This paper presents an investigation on the behaviour of conventional inverted pendulum with an inertia disk in its free extreme. The system is actuated by means of torques applied to the disk by a DC motor, mounted on the pendulum's arm. Thus, the system is underactuated since the pendulum can rotate freely around its pivot point. The dynamical model is given with three ordinary nonlinear differential equations. Using Poincare-Andronov-Hopf's theory, we find a new analytical formula for the first Lyapunov's value at the boundary of stability. It enables one to study in detail the bifurcation behaviour of the above dynamic system. We check the validity of our analytical results on the first Lyapunov's value by numerical simulations. Hence, we find some new results.

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Section: Problem Formulationmentioning

confidence: 99%

“…when the disk velocity is zero. Thus, according to [21], we consider an auxiliary control in the form:…”

Section: Problem Formulationmentioning

confidence: 99%

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

Николов¹,

Nedev²

2016

Journal of Theoretical and Applied Mechanics

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“…Other more complex control strategies reveal the great interest of the inverted pendulum in the field of control, as it is the case of control strategies based on space-state methods [9][10][11], control stabilization around homoclinic orbits [12], energy methods [13], passivity control [14] and bounded control [15] among others. However, the use of a simple control law to obtain chaotic behavior has been less used.…”

Section: Introductionmentioning

confidence: 99%

“…1 shows the layout of the pendulum system as well as the notation used to deduce the Lagrangian of the system. The pendulum is modeled by a mass m hanging at the end of a rod of negligible mass and length l, which is fixed to a support O [4][5][6], [7][8][9][10][11][12][13][14][15][16]. Let O'XY be an inertial frame and …”

Section: Introductionmentioning

confidence: 99%

Stability and chaotic behavior of a PID controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory

Pérez-Polo

Molina

Chica

et al. 2014

Applied Mathematics and Computation

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In this paper we investigate the stability and the onset of chaotic oscillations around the pointing-up position for a simple inverted pendulum that is driven by a control torque and is harmonically excited in the vertical and horizontal directions. The driven control torque is defined as a proportional plus integral plus derivative (PID) control of the deviation angle with respect to the pointing-down equilibrium position.The parameters of the PID controller are tuned by using the Routh criterion to obtain a stable weak focus around the pointing-up position, whose stability is investigated by using the normal form theory. The normal form theory is also used to deduce a simplified mathematical model that can be resolved analytically and compared with the numerical simulation of the complete mathematical model. From the harmonic prescribed motions for the pendulum base, necessary conditions for chaotic motion are deduced by means of the Melnikov function. When the pendulum is close to the unstable pointing-up position, the PID parameters are changed and the chaotic motion is destroyed, which is achieved by employing very small control signals even in the presence of random noise. The results of the analytical calculations are verified by full numerical simulations.

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“…Alonso et al [Alonso et al, 2005] used the bifurcation theory to classify different dynamical behaviors arising in the IWIP sub-20 ject to bounded continuous state feedback control law. To limit the maximum amplitude of the control action, the control law is subject to a smooth saturation function, which was first introduced in [Alonso et al, 2002] in order to stabilize the IWIP at the inverted position. The authors showed that the global dynamics of the IWIP change from a stable equilibrium point to a stable limit cycle via a Hopf bifurcation as certain control gains change.…”

mentioning

confidence: 99%

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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Controlling an Inverted Pendulum with Bounded Controls

Cited by 14 publications

References 15 publications

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

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

Stability and chaotic behavior of a PID controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory

From Hopf Bifurcation to Limit Cycles Control in Underactuated Mechanical Systems

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