Further Results on the Control Law via the Convex Hull of Ellipsoids
A new Lyapunov function based on the convex hull of ellipsoids was introduced in  for the study of uncertain and/or time-varying linear discrete-time systems with/without constraints. The new Lyapunov function has many attractive features such as: i) it provides a necessary and sufficient conditions for robust stability and robust stabilization; ii) the design conditions are formulated as linear matrix inequality constraints. The control law is obtained by solving a convex optimization problem online. This optimization problem generally does not have a closed-form solution, and hence it is solved by numerical methods. In this paper, we intend to complement the results in  by analyzing the geometric structures of the solution of the optimization problem, and of the control law. In particular, we show that the control law is a piecewise linear and continuous function of the state.
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