Quality-by-design by using the skewed spherical structured singular value

One of the tenets of Quality-by-Design (QbD) initiatives is the determination of a set of input parameters, called a design space, that ensures that the system outputs lie within a set of design specifications. Existing methods for the construction of design spaces for nonlinear systems are described by box constraints on the input parameters, which greatly limit the size of the design space specifications. This observation motivates the definition of the skewed spherical structured singular value, which is a generalization of both the skewed structured singular value and the spherical structured singular value. The scaled main loop theorem for the skewed spherical structured singular value is proved, which involves somewhat different steps due to the presence of the Frobenius norm. This theorem is used to derive a numerical algorithm that characterizes the ellipsoidal set of allowable real parametric uncertainties that achieve the output specifications for linear fractional systems.