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An Iterative Approach to the Fixed-Order Robust H∞ Control Problem Using a Sequence of Infeasible Controllers

It is well known that the robust disturbance attenuation against uncertainties can be achieved by the robust H controllers and some practical situations make us use the fixedorder controllers. These facts imply that the fixed-order robust H controllers are important for practical control problems. However it is difficult to design such robust controllers, because the robust H control problems include an infinite number of matrix inequality constraints, in other words, they are described by Robust Semi-Definite Programming (RSDP) problems. For obtaining a feasible solution of the RSDP problems coming from the robust control problems with state feedback controllers or full-order controllers, many numerical methods have been proposed. Classically, the quadratic stability theory, i.e. a common constant Lyapunov function for the entire uncertain set is used for reducing the infinite constraints to the finite ones at the expense of conservatism (Boyd et al. 1994). Recently, parameter dependent Lyapunov functions are used to improve the conservatism (Chesi et al. 2005) (Ichihara et al. 2003), (Kami et al. 2009) (Shaked 2001), (Xie 2008) and some one-shot type approaches using extended LMI conditions, which allows to use the affine parameter dependent Lyapunov functions, have been proposed (Pipeleers et al. 2009), (Shaked 2001), (Xie 2008). However these methods can not always produce the robust controller, because common additional variables are required and these methods can not be used for designing fixed-order controllers. In this sense, an iterative type approach may be useful to the problems such that these one-shot type approaches can not be applied. In the field of the numerical optimization, there are two types of iterative approaches for finding feasible or locally optimal solutions of the optimization problems: one is an interiorpoint approach which needs an initial feasible solution to be carried out and the other is an exterior-point approach which does not need it. From these facts, exterior-point approach can be efficient for obtaining the solutions of the problems such that feasible solutions are difficult to be found. However, there are no exterior-point approaches except those in (Iwasaki & Skelton 1995), (Kami & Nobuyama 2004), (Kami et al. 2009), (Vanbierviet 2009) for control problems to our knowledge.