Uncertainty Models and Robust Complex-Rational Controller Design for Flexible Structures

Abstract

Realparametricuncertaintiesin themodal dampingratiosandfrequenciesofe exiblestructuresarerepresented by complex uncertainties that can lead to robust controller designs satisfying robust performance specie cations. Thesecomplex uncertaintyblocksareusefulina π-synthesiscontrollerdesignprocedure.Twomodelsareproposed for modal parameter uncertainties. The e rst model uses a coprime factorization representation of the perturbed plant, whereas in the second model, a diagonal representation with complex eigenvalues is used. The innovation in the second method proposed is the use of a complex-rational controller design strategy, which offers tight uncertainty bounds and leads to a robust performance controller. The frequency response of the complex-rational controller is then approximated by a real-rational controller achieving the robust performance specie cations. LEXIBLE structures are generally characterized by the un- dampednaturalfrequenciesand dampingratiosoftheire exible modes. These parameters are subject to errors when they are esti- mated. Such uncertainties are important and should be taken into account in a robust controller design. The proper capture of modal parameter uncertainties in dynamic models of e exible structures for robust control has been the subject of ongoing efforts. Previ- ous research 1;2 used additive or multiplicative uncertainty models to take into account the variation in the dynamics of the plant. An- other way is to use certain heuristics to facilitate the representation of the parametric uncertainties in the e exible modes by a para- metric model. 3 These heuristics represent approximations in the parameter variation that are not generally realistic and lead to con- servative controller designs, that is, designs that cannot provide the desired performance in the face of realistic levels of parametric uncertainty. Recently, a model representing parametric uncertainties in the modes of a e exible structure was discussed in Ref. 4. Note that such models were developed 5 a few years ago. In Ref. 5, a model of dynamic uncertainty covering parametric variations in the e exible modes of a e exible structure was developed. This dynamic uncer- tainty has the virtue of being nonconservative, but only when the frequencies of the e exible modes are close to each other. In this paper, we propose to represent perturbations in the frequency and damping ratio of each e exible mode by a tight low-order dynamic uncertainty. Thus, we reduce the order of the augmented plant by half and transform the mixed real/complex uncertainty robust per- formanceπ-synthesis problem into an easier complex π synthesis. We use two techniques: The e rst is based on the coprime factor- ization framework, 6 and the second uses a complex diagonal modal representation to model the dynamics of the e exible structure and