Control of single input Hamiltonian systems based on the flatness of their tangent linearization

We explore the incremental flatness based control of single input under-actuated nonlinear Hamiltonian systems exhibiting a controllable tangent linearization model around a given equilibrium point. General properties of controllable linearized Hamiltonian systems are presented, which significantly ease the stabilizing, or output reference trajectory tracking, feedback controller design for the nonlinear system. Controllability of the tangent linear system is equivalent to its flatness. A flat filter controller and a set of nested controllers are presented which are based on incremental position measurements alone. There will be no need for explicit estimations of the conjugate momenta through observers. The proposed controller is constituted by a set of nested, second order linear Flat Filter compensation networks acting on the second order pure integration models, naturally present in the input-to-flat output system structure. For pure integration perturbed systems, Flat Filters have been shown to be equivalent, in general, to reduced order extended state observer based Active Disturbance Rejection controllers. A challenging stabilization problem for an unstable, nonlinear, two spring-mass system carrying an inverted pendulum is considered. Computer simulations illustrate the effectiveness of the proposed controller design in the presence of flat output measurement noise.