We present a scalable set-valued safety-preserving controller for constrained continuous-time linear time-invariant (LTI) systems subject to additive, unknown but bounded disturbance or uncertainty. The approach relies upon a conservative approximation of the discriminating kernel using robust maximal reachable sets---an extension of our earlier work on computation of the viability kernel for high-dimensional systems. Based on ellipsoidal techniques for reachability, a piecewise ellipsoidal algorithm with polynomial complexity is described that under-approximates the discriminating kernel under LTI dynamics. This precomputed piecewise ellipsoidal set is then used online to synthesize a permissive state-feedback safety-preserving controller. The controller is modeled as a hybrid automaton and can be formulated such that under certain conditions the resulting control signal is continuous across its transitions. We show the performance of the controller on a twelve-dimensional flight envelope protection problem for a quadrotor with actuation saturation and unknown wind disturbances.
We propose an algorithm which performs a progressive approximation of a viability kernel, iteratively using a classification method. We establish the mathematical conditions that the classification method should fulfil to guarantee the convergence to the actual viability kernel. We study more particularly the use of support vector machines (SVMs) as classification techniques. We show that they make possible to use gradient optimisation techniques to find a viable control at each time step, and over several time steps. This allows us to avoid the exponential growth of the computing time with the dimension of the control space. It also provides simple and efficient control procedures. We illustrate the method with some examples inspired from ecology