Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system

Symbiosis of RFPT-Based Adaptivity and the Modified Adaptive Inverse Dynamics Controller

The use of Lyapunov\u2019s \u201cdirect\u201d method for designing globally asymptotically stable controllers generates numerous, practically disadvantageous restrictions. The \u201cAdaptive Inverse Dynamic Controller for Robots (AIDCR)\u201d therefore suffers from various difficulties. As alternative design approach the \u201cRobust Fixed Point Transformations (RFPT)\u201d were introduced that instead of parameter tuning adaptively deforms the control signals computed by the use of a fixed approximate system model by observing the behaviour of the controlled system. It cannot guarantee global asymptotic stability but it is robust to the simultaneous presence of the unknown external disturbances and modelling imprecisions. In the paper it is shown that the RFPT-based design can co-operate with a modified version of the AIDCR controller in the control of \u201cMultiple Input-Multiple Output (MIMO)\u201d Systems. On the basis of certain function approximation theorems it is expected that this symbiosis works well in a wider class of physical systems than robots