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 both relative equilibria and ordinary equilibria of discrete mechanical systems using the controlled‑Lagrangian and matching methodology. While the continuous‑time theory relies on modifying the kinetic energy metric and adding potential‑energy shaping terms to achieve “matching” between the original and controlled dynamics, the discrete setting introduces new challenges: the discrete Euler–Lagrange equations contain explicit time‑step (Δt) dependence, and naïve discretisation of the continuous controlled Lagrangian typically leads to energy drift and a loss of the exact matching conditions. To overcome these obstacles the authors propose a two‑stage matching procedure.

First, kinetic matching is achieved by introducing two adjustable parameters: a scaling factor β that modifies the mass matrix and a novel scalar τ that compensates for the discretisation error in the momentum map. By selecting an appropriate momentum level p* (the discrete analogue of a conserved momentum value) or by tuning τ, the discrete momentum equation of the controlled system can be made identical to that of the original system, thereby preserving the reduced dynamics on the symmetry‑reduced space.

Second, potential matching is addressed by adding a Δt‑scaled gradient term to the shape equation of the controlled Lagrangian. This term corrects the discrepancy between the continuous potential energy and its discrete counterpart, ensuring that the controlled shape dynamics satisfy the same equilibrium conditions as the original system. The combined kinetic‑potential matching yields a discrete controlled Lagrangian whose Euler–Lagrange equations exactly reproduce the desired closed‑loop dynamics.

The theoretical development is illustrated with the classic cart‑pendulum on an incline. The system possesses a non‑trivial symmetry (translation of the cart) and a broken symmetry due to the incline angle, making it an ideal test case. The authors derive the discrete equations of motion, apply the two‑stage matching, and compare two implementation strategies: (i) fixing the momentum level p* and solving for β, and (ii) fixing β and solving for τ. Numerical simulations show that both strategies successfully stabilize the upright pendulum while keeping the cart at a fixed position, but the τ‑based approach exhibits greater robustness to variations in the sampling period. Energy plots confirm that the τ‑augmented controlled Lagrangian eliminates the drift that would otherwise appear in a straightforward discretisation.

Beyond the specific example, the paper discusses how the discrete controlled‑Lagrangian framework integrates naturally with digital controllers and model‑predictive control (MPC). Because the matching conditions are expressed directly in terms of the discrete time step, the designer can incorporate sampling‑rate constraints, actuator delays, and real‑time optimization of β and τ within an MPC horizon. This opens the door to applying the method to a wide range of real‑time robotic and aerospace systems where discrete‑time implementation is unavoidable.

In summary, the contribution of the paper is threefold: (1) identification of new phenomena that arise only in the discrete setting, notably the need for momentum‑level selection or an extra τ‑parameter; (2) formulation of a complete kinetic‑potential matching scheme that restores exact equivalence between original and controlled discrete dynamics; and (3) demonstration of practical applicability through a detailed cart‑pendulum example and discussion of digital/MPC extensions. The results bridge the gap between continuous‑time controlled‑Lagrangian theory and its implementation on digital hardware, providing a rigorous yet implementable pathway for stabilizing discrete mechanical systems with symmetry.