The Inverse Problem of the Calculus of Variations and the Stabilization of Controlled Lagrangian Systems

Abstract

We apply methods of the so-called 'inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the inverted pendulum on a cart and the inertia wheel pendulum. By making use of a condition that follows from Douglas' classification, we derive feedback controls for which the control system is variational. We then use the energy of a suitable controlled Lagrangian to provide a stability criterion for the equilibrium.

An Extension to the Theory of Controlled Lagrangians Using the Helmholtz Conditions

“…We now fix the parameters of the system to be m = 0.14 kg, M = 0.44 kg and l = 0.215 m as in [8] and take the initial conditions to be φ(0) = π/2 − 0.2 rad,φ(0) = 0.1 rad/s, s(0) = 0 m, anḋ s(0) = −3 m/s, also as in [8]. Below there is a matlab simulation of this situation with k = 35: This is the same case as the controlled systems that appear in [8] and [13] for the example of the inverted pendulum on a cart.…”

“…The system (27)-(28) fits into the class of systems dealt with in[13] and belongs to Case IIa1 from Douglas' classification since Φ 2 dγ 2(β 2 − αγ) cos(x) αγ − β 2 cos 2 (x) − 2βγk(αγ + β 2 cos 2 (x)) = 0 .…”

Robust output‐feedback control for the cart–pole system: a coupled super‐twisting sliding‐mode approach

“…For instance, in [6], a reinforcement learning algorithm is employed, although the stability is not studied. In contrast, an energy shaping approach is proposed in [7] to stabilise the system. In [8], an integral back‐stepping approach alongside a K filter is used to control a wheeled inverted pendulum whose model is considered to be the same as the cart–pole system.…”