ℋ2 dynamic output feedback scheduled controllers for linear time-varying polytopic systems: A convex LMI approach

This paper provides a convex condition to design dynamic output feedback scheduled controllers which ensure the closed-loop stability and minimize an upper bound to the H
₂
norm for linear systems whose matrices are affected by arbitrarily time-varying parameters belonging to a polytope. Differently from the conditions for the design of robust H
₂
dynamic controllers, which are nonconvex, the proposed design is entirely based on a convex linear matrix inequality optimization procedure. Moreover, in this paper, all the system matrices are affected by the vector of time-varying parameters which can vary arbitrarily fast inside the polytope. By means of variable elimination and also by exploiting the positivity of the parameters, it is shown that the design problem can be expressed as a convex optimization problem subject to a finite number of linear matrix inequality constraints formulated only in terms of the vertices of the polytope, avoiding the use of exhaustive gridding in the parameter space to compute a family of controllers. Numerical examples, including an application to the control of a model of a helicopter subject to abrupt failures of actuator, illustrate the efficiency of the proposed approach.