Generally, large-scale dynamical systems pose tremendous computational difficulties when applied in numerical simulations. In order to overcome these challenges we use several model reduction techniques. For stable linear models these techniques work very well and provide good approximations for the full model. However, large-scale unstable systems arise in many applications. Many of the known model reduction methods are not very robust, or in some cases, may not even work if we are dealing with unstable systems. When approximating an unstable system by a reduced order model, accuracy is not the only concern. We also need to consider the structure of the reduced order model. Often, it is important that the number of unstable poles in the reduced system is the same as the number of unstable poles in the original system. The Iterative Rational Krylov Algorithm (IRKA) is a robust model reduction technique which is used to locally reduce stable linear dynamical systems optimally in the ℋ₂-norm. While we cannot guarantee that IRKA reduces an unstable model optimally, there are no numerical obstacles to the reduction of an unstable model via IRKA. In this thesis, we investigate IRKA's behavior when it is used to reduce unstable models. We also consider systems for which we cannot obtain a first order realization of the transfer function. We can use Realization-independent IRKA to obtain a reduced order model which does not preserve the structure of the original model. In this paper, we implement a structure preserving algorithm for systems with nonlinear frequency dependency. / Master of Science

Research Doctorate - Doctor of Philosophy (PhD) / This dissertation presents some new approaches to addressing the main issues encountered by practitioners in the implementation of linear model predictive control(MPC), namely, stability, feasibility, complexity and the size of the region of attraction. When stability guaranteeing techniques are applied nominal feasibility is also guaranteed. The most common technique for guaranteeing stability is to apply a special weighting to the terminal state of the MPC formulation and to constrain the state to a terminal region where certain properties hold. However, the combination of terminal state constraints and the complexity of the MPC algorithm result in regions of attraction that are relatively small. Small regions of attraction are a major problem for practitioners. The main approaches used to address this issue are either via the reduction of complexity or the enlargement of the terminal region. Although the ultimate goal is to enlarge the region of attraction, none of these techniques explicitly consider the upper bound of this region. Ideally the goal is to achieve the largest possible region of attraction which for constrained systems is the null controllable set. For the case of systems with a single unstable pole or a single non-minimum phase zero their null controllable sets are defined by simple bounds which can be thought of as implicit constraints. We show in this thesis that adding implicit constraints to MPC can produce maximally controllable systems, that is, systems whose region of attraction is the null controllable set. For higher dimensional open-loop unstable systems with more than one real unstable mode, the null controllable sets belong to a class of polytopes called zonotopes. In this thesis, the properties of these highly structured polytopes are used to implement a new variant of MPC, which we term reduced parameterisation MPC (RP MPC). The proposed new strategy dynamically determines a set of contractive positively invariant sets that require only a small number of parameters for the optimisation problem posed by MPC. The worst case complexity of the RP MPC strategy is polylogarithmic with respect to the prediction horizon. This outperforms the most efficient on-line implementations of MPC which have a worst case complexity that is linear in the horizon. Hence, the reduced complexity allows the resulting closed-loop system to have a region of attraction approaching the null controllable set and thus the goal of maximal controllability.

Unstable Systems or Why Is My Junk So Raw? is an exploration in the raw aesthetics of exposed electronics; showing the complicated systems that make our everyday electronics work using the visual language of formalism to display these “broken” consumer electronics as art. The work in my thesis show explores the creative potential of death and impermanence through the failing of technology. The work in the exhibition combines my interest and childhood fascination in electronics as well as my experience with my father’s illness. Accidentally and intentionally broken TV’s and electronics are producing live glitches which emphasize the instability of these otherwise closed systems.