Energy shaping control of an inverted flexible pendulum fixed to a cart

Abstract

Control of compliant mechanical systems is increasingly being researched for several applications including flexible link robots and ultra-precision positioning systems. The control problem in these systems is challenging, especially with gravity coupling and large deformations, because of inherent underactuation and the combination of lumped and distributed parameters of a nonlinear system. In this paper we consider an ultra-flexible inverted pendulum on a cart and propose a new nonlinear energy shaping controller to keep the pendulum at the upward position with the cart stopped at a desired location. The design is based on a model, obtained via the constrained Lagrange formulation, which previously has been validated experimentally. The controller design consists of a partial feedback linearization step followed by a standard PID controller acting on two passive outputs. Boundedness of all signals and (local) asymptotic stability of the desired equilibrium is theoretically established. Simulations and experimental evidence assess the performance of the proposed controller.

Robust balancing control of flexible inverted-pendulum systems

“…Also, is the input matrix which, without loss of generality, is assumed of the form = [ ; 0 ( − )× ] so that rank( ) = < , and ∈ ℝ is the control input, while the term ∈ ℝ represents the disturbance. Similarly to [22,25], the proposed control law employs the full state (i.e. ,̇ are measurable).…”

“…= 0). The control law (10) for system (4) consists of the nonlinear PID [25] augmented by the disturbance-compensation term * , and the adaptation law (8). Since does not directly depend on we have = ( * − ) according to Proposition 1.…”

“…Nevertheless, both approaches result in a more elaborated controller design and in a relatively large number of parameters. Recently, an alternative energy-shaping design that obviates the solution of PDE while maintaining a more intuitive structure akin to a PID was proposed in [22][23][24] and applied to a flexible inverted-pendulum in [25], although external disturbances were disregarded for simplicity. While recent works proposed robustness enhancement strategies for energy-shaping control [26][27][28], most research has been focusing on matched disturbances (i.e.…”

Analysis of the nonlinear chaotic behavior of an inverted flexible pendulum system affected by impulses

“…The author developed a dynamic model to predict multiple equilibria. Recent studies developed [3] control approaches to achieve one equilibrium. Impulsive control that allows the stabilization and synchronization of chaotic systems using only small control impulses was applied by [4].…”

Periodic Responses of a Rotating Hub-Beam System with a Tip Mass under Gravity Loads by the Incremental Harmonic Balance Method

“…As for a flexible hub geometrical nonlinearity beam with a tip mass, Emam [18] employed a flexural model to study the dynamic responses of a flexible hub geometrical beam with a tip mass of which the hub is restrained by a translational and a rotational spring; such a model accounts for the geometrical coupling between the axial and lateral deformations. If gravity of the beam is much less than the gravity of tip mass, it is not necessary to consider the effect of the gravity load of beam [19].…”