Sampled-data (SD) systems, which are composed of both discrete- and continuous-time components, are arguably one of the most common classes of cyberphysical systems in practice; most modern controllers are implemented on digital platforms while the plant dynamics that are being controlled evolve continuously in time. As with all cyberphysical systems, ensuring hard constraint satisfaction is key in the safe operation of SD systems. A powerful analytical tool for guaranteeing such constraint satisfaction is the viability kernel: the set of all initial conditions for which a safety-preserving control law (that is, a control law that satisfies all input and state constraints) exists. In this paper we present a novel sampling-based algorithm that tightly approximates the viability kernel for high-dimensional sampled-data linear time-invariant (LTI) systems. Unlike prior work in this area, our algorithm formally handles both the discrete and continuous characteristics of SD systems. We prove the correctness and convergence of our approximation technique, provide discussions on heuristic methods to optimally bias the sampling process, and demonstrate the results on a twelve-dimensional flight envelope protection problem.
 Every convex polytope is both the intersection of a finite set of halfspaces and the convex hull of a finite vertex set. Transforming from the halfspaces (vertices, respectively) to the vertices (halfspaces, respectively) is called vertex enumeration (facet enumeration, respectively). It is an open problem whether there is an algorithm for these two problems polynomial in the input and the output size. For each of the known methods, this thesis develops a characterization of what constitutes an easy or difficult input. Example families of polytopes are presented that show that none of the known methods will yield a polynomial algorithm. On the other hand, a family of polytopes difficult for one class of algorithms can (sometimes) be easily solvable for another class of algorithms; the characterizations given here can be used to guide a choice of algorithms. Similarly, although the general problems of vertex and facet enumeration are equivalent by the duality of convex polytopes, for fixed polytope family and algorithm, one of these directions can be much easier than the other. This thesis presents a new class of algorithms that use the easy direction as an oracle to solve the seemingly difficult direction. This paper addresses the attraction domain of model-based predictive control (MPC) for nonlinear systems with control input and state constraints. Based on a stability condition of nonlinear MPC, a method to determine the terminal weighting term in the performance index and the terminal stabilising control law to maximise the domain of attraction of the nonlinear MPC is proposed. The problem of maximisation of the attraction region is recast as a well-defined optimisation problem. By an LMI based optimisation approach, the terminal weighting item and fictitious terminal stabilising control law axe optimised to enlarge the attraction domain and hence the feasibility domain of the nonlinear MPC method. The proposed method is illustrated by a numerical example and favourably compared with existing results. This paper presents a new (geometrical) approach to the computation of polyhedral (robustly) positively invariant (PI) sets for general (possibly discontinuous) nonlinear discrete-time systems possibly affected by disturbances. Given a @b-contractive ellipsoidal set E, the key idea is to construct a polyhedral set that lies between the ellipsoidal sets @bE and E. A proof that the resulting polyhedral set is contractive and thus, PI, is given, and a new algorithm is developed to construct the desired polyhedral set. The problem of computing polyhedral invariant sets is formulated as a number of quadratic programming (QP) problems. The number of QP problems is guaranteed to be finite and therefore, the algorithm has finite termination. An important application of the proposed algorithm is the computation of polyhedral terminal constraint sets for model predictive control based on quadratic costs.