two outputs, the error signals z that we want to minimize, and the measured variables v, that we use to control the system. v is used in K to calculate the manipulated variables u. Notice that all these are generally vectors, whereas P and K are matrices. In formulae, the system is: It is therefore possible to express the dependency of z on w as: Called the lower linear fractional transformation, is defined (the subscript comes from lower): Therefore, the objective of control design is to find a controller such that is minimised according to the norm. The same definition applies to control design. The infinity norm of the transfer function matrix is defined as: where is the maximum singular value of the matrix . The achievable H∞ norm of the closed loop system is mainly given through the matrix D11 (when the system P is given in the form (A, B1, B2, C1, C2, D11, D12, D22, D21)). There are several ways to come to an H∞ controller: A Youla-Kucera parametrization of the closed loop often leads to very high-order controller. Riccati-based approaches solve two Riccati equations to find the controller, but require several simplifying assumptions. An optimization-based reformulation of the Riccati equation uses linear matrix inequalities and requires fewer assumptions. See also Hardy space H square H-infinity loop-shaping Linear-quadratic-Gaussian control (LQG) Rosenbrock system matrix References Bibliography . . . . . . Control theory Hardy spaces