Synthesis of anisotropic suboptimal controllers via convex optimization

This paper considers a disturbance attenuation problem for a linear discrete time invariant system under random disturbances with imprecisely known distributions. The statistical uncertainty is measured in terms of the mean anisotropy functional. The disturbance attenuation capabilities of the system are quantified by the anisotropic norm which is a stochastic counterpart of the H∞ norm. The designed anisotropic suboptimal controller generally is a dynamic fixed-order output-feedback compensator which is required to stabilize the closed-loop system and keep its anisotropic norm below a prescribed threshold value. Rather than resulting in a unique controller, the suboptimal design procedure yields a family of controllers, thus providing freedom to impose some additional performance specifications on the closed-loop system. The general fixed-order synthesis procedure employs solving a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities in reciprocal matrices which make the general optimization problem nonconvex. By applying the known standard convexification procedures it is shown that the resulting optimization problem is convex for the full-information state-feedback, output-feedback full-order controllers, and static outputfeedback controller for some specific classes of plants defined by certain structural properties. In the convex cases, the anisotropic γ-optimal controllers are obtained by minimizing the squared norm threshold value subject to convex constraints. In a sense, the anisotropic controller seems to offer a promising and flexible trade-off between H2 and H∞ controllers which are its limiting cases. In comparison with the state-space solution to the anisotropic optimal controller synthesis problem presented before which results in a unique full-order estimator-based controller defined by a complex system of cross-coupled nonlinear matrix algebraic equations, the proposed optimization-based approach is novel and does not require developing specific homotopy-like computational algorithms.