Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations

Abstract

Control of underactuated mechanical systems via energy shaping is a well-established, robust design technique. Unfortunately, its application is often stymied by the need to solve partial differential equations (PDEs). In this paper a new, fully constructive, procedure to shape the energy for a class of mechanical systems that obviates the solution of PDEs is proposed. The control law consists of a first stage of partial feedback linearization followed by a simple proportional plus integral controller acting on two new passive outputs. The class of systems for which the procedure is applicable is identified imposing some (directly verifiable) conditions on the systems inertia matrix and its potential energy function. It is shown that these conditions are satisfied by three benchmark examples. Index Terms-Nonlinear systems, passivity-based control, mechanical systems, stabilization.0018-9286 (c)

Global stabilisation of underactuated mechanical systems via PID passivity-based control

“…Second, the observation that it is possible to construct a Lyapunov function for the desired equilibrium via a suitable choice of the aforementioned weights and the PID gains and initial conditions. The results reported here follow the same research line as [7] and [20]-bridging the gap between the Hamiltonian and the Lagrangian formulations used, correspondingly, in these papers. Two additional improvements to our previous works are the removal of a non-robust cancellation of a potential energy term and the establishment of equilibrium attractivity under weaker assumptions.…”

“…The derivations in [20] are done working with the Hamiltonian representation of the system, and the main modification is the introduction of a suitable momenta coordinate change that directly reveals the new passive outputs. These passive outputs are different from the ones obtained in [7] using the Lagrangian formalism. Therefore, it is not possible to compare the realms of applicability of both methods-see Remark 6 in [20].…”

“…Second, adding this function to the systems storage function, to generate a bona fide Lyapunov function by assigning a minimum at the aforementioned equilibrium. It is shown in [7] that many benchmark examples satisfy the (easily verifiable) conditions on the systems inertia matrix and its potential energy function imposed by the method.…”

“…A key feature of total energy shaping methods is that the mechanical structure of the system is preserved in closed-loop, a condition that gives rise to the aforementioned PDE, which characterise the assignable energy functions. In [7] it was recently proposed to relax this constraint, and concentrate our attention on the energy shaping objective only. That is, we look for a control law that stabilizes the desired equilibrium assigning to the closed-loop a Lyapunov function of the same form as the energy function of the open-loop system but with new, desired inertia matrix and potential energy function.…”

“…It is widely recognised that feedback linearization, which involves the exact cancelation of nonlinear terms, is intrinsically non-robust. Interestingly, it has recently been shown in [20] that, for the class of systems considered in [7], it is possible to identify two new passive outputs without the feedback linearization step. The derivations in [20] are done working with the Hamiltonian representation of the system, and the main modification is the introduction of a suitable momenta coordinate change that directly reveals the new passive outputs.…”