Teixeira singularities in 3D switched feedback control systems

Abstract: This paper is concerned with the analysis of a singularity that can occur in threedimensional discontinuous feedback control systems. The singularity is the two-fold – a tangency of orbits to both sides of a switching manifold. Particular attention is placed on the Teixeira singularity, which previous literature suggests as a mechanism for dynamical transitions in this class of systems. We show that such a singularity cannot occur in classical single-input single-output systems in the Lur’e form. It is, however, a potentially dangerous phenomenon in multiple-input multiple-output switched control systems.The theoretical derivation is illustrated by means of a representative example

Dynamics and Control of Non-smooth Systems with Applications to Supercavitating Vehicles

The subject matter of this dissertation relates to the dynamics of non-smooth vehicle systems, and in particular, supercavitating vehicles. These high-speed underwater vehicles are designed to have sustained vaporous or ventilated gas cavities that form over the entire vehicle. In terms of the modeling, the system non-smoothness is caused by the interaction forces generated when the vehicle contacts the cavity. These planing interactions can cause stable and unstable dynamics, some of which could be limit-cycle dynamics. Here, planing forces are considered on the basis of non-cylindrical cavity shapes that include shifts induced by the cavitator angle of attack. Incorporating these realistic physical effects into a vehicle system model generates a unique hydrodynamic non-smoothness that is characterized by non-constant switching boundaries and non-constant switched dynamics. Nonlinear stability analyses are carried out, Hopf bifurcations of equilibrium solutions are identified, and stabilizing control is investigated. Also considered is partially cavitating system dynamics, where active fin forces are used to support the vehicle. Non-steady planing is also considered, which accounts for vehicle motions into the cavity, and this planing provides a damping-like component in the planing force formulation. Modeled with non-steady planing is a physical time delay relating to the fact that the cavity, where planing occurs, is based on the previous cavitator position and orientation data. This delay is found to be stabilizing for certain values of speed. Maneuvering is considered by using inner-loop and outer-loop control schemes. A feedback inner-loop scheme helps reject fast planing instabilities, while a numeric optimal control approach is used to generate outer-loop commands to guide the vehicle through desired maneuvers. The maneuvers are considered for operations with tight body to cavity clearance, and in which planing is prevalent. Simple search algorithms along with a penalty method for handling the constraints are found to work the best due to the complexity of the non-smooth system dynamics

Nonlinear dynamics of DC-DC converters

Power electronic converters are time-varying, nonlinear dynamical systems. They exhibit a wide range of steady-state responses. The desired behaviour is a stable periodic motion around a predefined value with a frequency that is equal to that of the external clock. However, as parameters vary the operation can lose stability and go from one regime to another. Such phenomena are termed bifurcations and can degrade the output performance of the converter. Hence, it is of practical importance to know the conditions that cause such bifurcations to occur and to design the system so that it operates in the desired region. In the past, engineers have typically analysed the stability of power electronic systems by linearising the model about a fixed point. This captures the low-frequency properties while ignoring the detailed dynamics occurring at frequencies higher than the external clock. However, the demand for better functionality, reliability and performance means an in-depth analysis into the complex behaviour exhibited by dc-dc converters is required. Traditionally, dc-dc converters are employed with analog controllers whose function is to regulate the circuit. With advances in technology, digital control has become a potentially advantageous alternative to analog control. One of the main advantages of digital control is the ability to design more sophisticated design strategies to enable high performance dc-dc converters e.g. digital state-feedback control. Unfortunately, little work exists in the area of the effect of noise on digital control. This is a field that requires intensive study as to completely understand the nonlinear dynamics so as to enable accurate and economic designs. The aim of this thesis is to address these issues through the application of advanced nonlinear mathematics. The stability of power electronic systems is assessed with a view to developing design guidelines in order to ensure stable operation over a wide operating region

Optimal economic planning and control for the management of ecosystems

In recent years the interest on sustainable systems has increased significantly. Among the many interested problems, creating and restoring sustainable ecosystems is a challenging and complex problem. One of the fundamental problems within this area is the imbalance between species that have a predator-prey relationship. Solutions involving management have become an integral player in many environments. Management systems typically use ad hoc methods to develop harvesting policies to control the populations of species to desired numbers. In order to amalgamate intelligence and structure, ecological systems require a diverse research effort from three primary fields: ecology, economics, and control theory. In this thesis, all three primary fields aforementioned are researched to develop a theoretical framework that includes an optimal trajectory planning system that exploits an ecosystem to maximize profits for the supporting community, and a robust control system design to track the optimal trajectories subjected to exogenous disturbances. Population ecology is used to select a model that identifies the key characteristics a management system needs to understand the behavior of the natural environment. A bioeconomic model is developed to relate the species populations to revenue. The nonlinear ecosystem is transformed into a linear parameter-varying (LPV) system that is then controlled using hinf synthesis and the gain scheduling methodology. The consequences of the results in this thesis are that optimal trajectories of an ecosystem can be obtained by constructing and solving a nonlinear programming problem (NLP), and the LPV based gain scheduling approach produces a robust controller that rejects disturbances and advises quality control policies to the manager an ecosystem. The LPV controller achieves comparable profits with satisfactory tracking performance while minding the induced costs of its high frequency output. Implications of constraining the control effort when designing for robustness are observed. Overall, the theoretical framework provides a solid foundation for future research on the understanding and improvement of ecosystem management

Uniform finite time stabilisation of non-smooth and variable structure systems with resets

This thesis studies uniform finite time stabilisation of uncertain variable structure and non-smooth systems with resets. Control of unilaterally constrained systems is a challenging area that requires an understanding of the underlying mechanics that give rise to reset or jumps while synthesizing stabilizing controllers. Discontinuous systems with resets are studied in various disciplines. Resets in states are hard nonlinearities. This thesis bridges non-smooth Lyapunov analysis, the quasi-homogeneity of differential inclusions and uniform finite time stability for a class of impact mechanical systems. Robust control synthesis based on second order sliding mode is undertaken in the presence of both impacts with finite accumulation time and persisting disturbances. Unlike existing work described in the literature, the Lyapunov analysis does not depend on the jumps in the state while also establishing proofs of uniform finite time stability. Orbital stabilization of fully actuated mechanical systems is established in the case of persisting impacts with an a priori guarantee of finite time convergence between t he periodic impacts. The distinguishing features of second order sliding mode controllers are their simplicity and robustness. Increasing research interest in the area has been complemented by recent advances in Lyapullov based frameworks which highlight the finite time Convergence property. This thesis computes the upper bound on the finite settling time of a second order sliding mode controller. Different to the latest advances in the area, a key contribution of this thesis is the theoretical proof of the fact that finite settling time of a second order sliding mode controller tends to zero when gains tend to infinity. This insight of the limiting behaviour forms the basis for solving the converse problem of finding an explicit a priori tuning formula for the gain parameters of the controller when and arbitrary finite settling time is given. These results play a central role ill the analysis of impact mechanical systems. Another key contribution of the thesis is that it extends the above results on variable structure systems with and without resets to non-smooth systems arising from continuous finite time controllers while proving uniform finite time stability. Finally, two applications are presented. The first application applies the above theoretical developments to the problem of orbital stabilization of a fully actuated seven link biped robot which is a nonlinear system with periodic impacts. The tuning of the controller gains leads to finite time convergence of the tracking errors between impacts while being robust to disturbances. The second application reports the outcome of an experiment with a continuous finite time controller