Modeling and Control of Linear Two-time Scale Systems: Applied to Single-Link Flexible Manipulator

This paper deals with the problem of H _∞ control of linear two-time scale systems. The authors’ attention is focused on the robust regulation of the system based on a new modeling approach under the assumption of norm-boundedness of the fast dynamics. In the proposed approach, the fast dynamics are treated as a norm-bounded disturbance (dynamic uncertainty). In this view, the synthesis is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. It should be noted, however, that this scheme is significantly different from the conventional approaches of order reduction for linear two-time scale systems. Specifically, in the present work, explicitly or implicitly, all the dynamics of the system are taken into consideration. In other words, the portion that is treated as a perturbation is incorporated in the design by its maximum possible gain – in the L ₂ sense – over different values of the inputs. One of the advantagesof this approach is that – unlike in the conventional approaches of the order reduction – the reduced-order system still keeps some information of the ‘deleted’ subsystem. Also, we consider the robust stability analysis and stability bound improvement of perturbed parameter (ɛ) in the two-time scale systems by using linear fractional transformations and structured singular values (μ) approach. In this direction, by introducing the parametric uncertainty and dynamic uncertainty in the two-time scale systems, we represent the system as a standard μ-interconnection framework by using linear fractional transformations, and derive a set of new stability conditions for the system in the frequency domain. The exact solution of ɛ-bound is characterized. It is shown that, in spite of the coupling between the dynamic uncertainties and certain dynamics, the designed H _∞ controller stabilizes the overall closed-loop system, in the presence of norm-bounded disturbances. To show the effectiveness of the approach, the modeling of the single-link flexible manipulator and control of the Tip-position of the manipulator by utilizing the mentioned method are presented in the case study.