Systems involving quantisation arise in many areas of engineering, especially when digital implementations are involved. In this thesis we consider different aspects of quantisation in feedback control systems. We study two topics of interest: (a) quantisers that quadratically stabilise a given system and are efficient in the use of their quantisation levels and (b) the derivation of ultimate bounds for perturbed systems, especially when the perturbations arise from the use of quantisers. In the first part of the thesis we address problem (a) above. We consider quadratic stabilisation of discrete-time multiple-input systems by means of quantised static feedback and we measure the efficiency of a quantiser via the concept of quantisation density. Intuitively, the lower the density of a quantiser is, the more separated its quantisation levels are. We thus deal with the problem of optimising density over all quantisers that quadratically stabilise a given system with respect to a given control Lyapunov function. Most of the available results on this problem treat single-input systems, and the ones that deal with the multiple-input case consider only two-input systems. In this thesis, we derive several new results for multiple-input systems and also provide an alternative approach to deal with the single-input case. Our new results for multiple-input systems include the derivation of the structure of optimal quantisers and the explicit design of multivariable quantisers with finite density that are able to quadratically stabilise systems having an arbitrary number of inputs. For single-input systems, we provide an alternative approach to the analysis and design of optimal quantisers by establishing a link between the separation of the quantisation levels of a quantiser and the size of its quantisation regions. In the second part of the thesis we address problem (b) above. In the presence of perturbations, asymptotic stabilisation may not be possible. However, there may exist a bounded region that contains the equilibrium point and has the property that the system trajectories converge to this bounded region. When this bounded region exists, we say that the system trajectories are ultimately bounded, and that this bounded region is an ultimate bound for the system. The size of the ultimate bound quantifies the performance of the system in steady state. Hence, it is important to derive ultimate bounds that are as tight as possible. This part of the thesis addresses the problem of ultimate bound computation in settings involving several scalar quantisers, each having different features. We consider each quantised variable in the system to be a perturbed copy of the corresponding unquantised variable. This turns the original quantised system into a perturbed system, where the perturbation has a natural \emph{componentwise} bound. Moreover, according to the type of quantiser employed, the perturbation bound may depend on the system state. Typical methods to estimate ultimate bounds are based on the use of Lyapunov functions and usually require a bound on the norm of the perturbation. Applying these methods in the setting considered here may disregard important information on the structure of the perturbation bound. We therefore derive ultimate bounds on the system states that explicitly take account of the componentwise structure of the perturbation bound. The ultimate bounds derived also have a componentwise form, and can be systematically computed without having to, e.g. select a suitable Lyapunov function for the system. The results of this part of the thesis, though motivated by quantised systems, apply to more general perturbations, not necessarily arising from quantisation. / PhD Doctorate

Graded ideals over the polynomial ring are studied deeply with a huge of methods and results. Over the exterior algebra, there are not much known about the structures of minimal graded resolutions, Gröbner fans of graded ideals or the Koszul property of algebras defined by graded ideals. We study componentwise linearity, linear resolutions of graded ideals as well as universally, initially and strongly Koszul properties of graded algebras defined by a graded ideals over the exterior algebra. After that, we apply our results to Orlik-Solomon ideals of hyperplane arrangements and show in which way the exterior algebra is useful in the study of related combinatorial objects.