Controlled Lagrangians and Stabilization of Discrete Mechanical Systems I

Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic matching procedure. Specifically, new terms in the controlled shape equation that are necessary for potential matching in the discrete setting are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline. The paper also discusses digital and model predictive controllers.

💡 Research Summary

The paper develops a systematic framework for stabilizing discrete mechanical systems using the method of controlled Lagrangians within the discrete variational mechanics setting. Starting from the continuous‑time theory, where a mechanical system’s Lagrangian is modified by kinetic shaping (velocity shifts) and potential shaping (energy modification) to embed a control law, the authors translate these ideas to the discrete domain by employing a discrete Lagrangian (L_d(q_k,q_{k+1})) that approximates the action integral over a time step (h). The discrete Euler–Lagrange equations derived from the discrete Hamilton principle, together with a discrete Lagrange–d’Alembert principle for external forces, provide the equations of motion for the uncontrolled system.

The core of the contribution lies in the matching procedure for discrete systems. For a system with one shape variable (\phi) and one group variable (s), the continuous‑time controlled Lagrangian (L_{\tau,\sigma}) (with a velocity shift (\tau(\phi)=\kappa\beta(\phi)) and a kinetic correction (\sigma)) is discretized to obtain (L_d^{\tau,\sigma}). The authors show that, unlike the continuous case where matching holds for any momentum level, the discrete matching imposes a relation between the chosen momentum level (\mu) and the actual discrete momentum (p): (\sigma = -\kappa/\gamma) and (\mu = p/(1+\gamma\kappa)). Consequently, the designer must either select a specific momentum level or introduce an additional parameter (\lambda) into the controlled Lagrangian, leading to a modified discrete Lagrangian ( \Lambda_d^{\tau,\sigma,\lambda}). With the choice (\lambda = -p), the matching conditions become independent of the momentum level, simplifying implementation.

The paper also addresses potential shaping for systems whose potential energy breaks the symmetry. In the discrete setting, the shape equation acquires non‑conservative forcing terms that must be compensated. The authors incorporate these terms into the controlled Lagrangian and derive the corresponding discrete matching conditions.

Stability analysis proceeds by linearizing the reduced discrete dynamics around the relative equilibrium (\phi=0). The discrete energy approximation is shown to be preserved, and spectral stability is guaranteed when the kinetic shaping parameter satisfies (\kappa > (\alpha\gamma - \beta^2(0))/(\beta^2(0)\gamma)). This condition mirrors the continuous‑time result, confirming consistency as (h\to0).

To illustrate the theory, the authors apply it to a cart‑pendulum system on an incline. They construct the discrete controlled Lagrangian, compute the required control input (u_k), and perform numerical simulations. The results demonstrate that the inverted pendulum configuration, which is unstable in the uncontrolled discrete system, becomes asymptotically stable when the discrete control law and optional dissipation‑emulating terms are added.

Beyond stabilization, the paper explores digital control and model predictive control (MPC). Because the discrete controlled dynamics are naturally expressed in a sampled‑data form, the control input can be held constant over each time step, facilitating digital implementation. Moreover, the discrete variational integrator provides accurate state predictions that are directly usable in an MPC scheme, allowing real‑time optimal control while preserving the geometric structure of the system.

In summary, the authors extend the controlled Lagrangian methodology to discrete mechanical systems, uncovering new phenomena such as momentum‑level dependence and the necessity of an extra parameter for kinetic matching. They provide rigorous matching conditions, stability proofs, and a concrete example, and they demonstrate how the discrete framework integrates seamlessly with digital and predictive control strategies. The work lays a solid foundation for future extensions to non‑abelian symmetries, nonholonomic constraints, and higher‑order variational integrators.