Robust positively invariant cylinders for constrained sliding mode control designs

We consider the problem of designing sliding mode controllers for linear systems subject to disturbance and constraints in the states and control magnitude. The use of infinite cylinders as primary, positively invariant sets is motivated by a coordinate transformation where the sliding motion is decoupled from the overall convergence to the origin. We give robust positive invariance conditions for cylinders having convex and compact cross sections. Robust positively invariant cylinders are intersected with the state constraints to yield sets which, under some conditions, retain the invariance and satisfy the constraints. For the case of cylinders with ellipsoidal cross sections, we provide a decision procedure that is used to qualify each state constraint during the design process. A numerical example for a third order plant illustrates the method.