Control theory

In this dissertation we present a computational approach to Conley's Decomposition
Theorem, which gives a global decomposition of dynamical systems, and
introduce an explicit numerical algorithm with computational complexity bounds
for computing global dynamical structures of a continous map including attractorrepeller
pairs and Conley's Lyapunov function. The approach is based on finite spatial
discretizations and combinatorial multivalued maps. The method is successful
in exhibiting approximations of attractor-repeller pairs, invariant sets, and Conley's
Lyapunov function. We used the C++ language to code the algorithm.

In this thesis, we study the input-to-state stability (scISS) property and related characterizations for discrete-time nonlinear systems. Variations of scISS property were employed in solving particular control problems. The main contribution of this work is to provide a detailed analysis on the relations among various types of notations related to system stability and show that most scISS results for continuous-time nonlinear system can be extended to discrete-time case.

In recent years robots have become increasingly important in many areas. Along with the development of robot-arm control theory, there has been an increased demand for faster and more reliable control systems. In this thesis, a parallel technique is applied to all of the units of a robot control system. Also, software fault-tolerance mechanisms such as timeout, conversation, exception handling, and their Occam implementations, are considered. A simulation study shows that pipelining, together with a multiprocessing system, increases the performance of this real-time system, and it is a convenient way to speed up robot controller execution. While we have not evaluated the increase in reliability, we have shown that these fault tolerance mechanisms can be conveniently implemented in this type of application.

In this thesis, a novel two-dimensional learning control scheme for robot manipulators is proposed. The convergence of the scheme for a general n-degree of freedom robot is shown. In the next part of the thesis, an algorithm for the approximation of a two-dimensional causal, recursive, separable-in-denominator (CRSD) filter, using the impulse response and autocorrelation data, is presented. The stability of the designed filter is discussed and it is shown that the approximated filter is always stable. The simulation results for the approximation technique as well as the two-dimensional learning control scheme are also included in the thesis.

This thesis presents several algorithms to treat the problem
of closed-loop near minimum-time controls with
discontinuities. First, a neighboring control algorithm is
developed to solve the problem in which controls are bounded
by constant constraints. Secondly, the scheme is extended
to account for state-dependent control constraints. And
finally, a path tracking algorithm for robotic manipulators
is presented, which is also a neighboring control algorithm.
These algorithms are suitable for real time controls because
the on-line computations involved are relatively simple.
Simulation results show that these algorithms work well
despite the fact that the prescribed final points can not be
reached exactly.

Model reduction of large-scale systems over a specified frequency range of operation is studied in this research and reported in this dissertation. Frequency-domain balanced structures with integration of singular perturbation are proposed for model reduction of large-scale continuous-time as well as discrete-time systems. This method is applied to both open-loop as well as closed-loop systems. It is shown that the response of reduced systems closely resemble that of full order systems within a specified frequency range of operation. Simulation experiments for the model reduction of several large-scale, continuous and discrete-time systems demonstrate the superiority of the proposed technique over the previously available methods.

In this work, we investigate input-to-state stability (ISS) and other related stability properties for control systems with time-delays. To overcome the complexity caused by the presence of the delays, we adopt a Razumikhin approach. The underlying idea of this approach is to treat the delayed variables as system uncertainties. The advantage of this approach is that one works in the more familiar territory of stability analysis for delay-free systems in the context of ISS instead of carrying out stability analysis on systems of functional differential equations. Our first step is to provide criteria on ISS and input-to-input stability properties based on the Razumikhin approach. We then turn our attention to large-scale interconnected systems. It has been well recognized that the small-gain theory is a powerful tool for stability analysis of interconnected systems. Using the Razumikhin approach, we develop small-gain theorems for interconnected systems consisting of two or more subs ystems with time-delays present either in the interconnection channels or within the subsystems themselves. As an interesting application, we apply our results to an existing model for hematopoesis, a blood cell production process,and improve the previous results derived by linear methods. Another important stability notion in the framework of ISS is the integral ISS (iISS) property. This is a weaker property than ISS, so it supplies to a larger class of systems. As in the case of ISS, we provide Razumikhin criteria on iISS for systems with delays. An example is presented to illustrate that though very useful in practice, the Razumikhin approach only provides sufficient conditions, not equivalent conditions. Finally, we address stability of time-varying systems with delays in the framework of ISS.

We generalize the theory of stochastic impulse control of jump diffusions introduced by Oksendal and Sulem (2004) with milder assumptions. In particular, we assume that the original process is affected by the interventions. We also generalize the optimal central bank intervention problem including market reaction introduced by Moreno (2007), allowing the exchange rate dynamic to follow a jump diffusion process. We furthermore generalize the approximation theory of stochastic impulse control problems by a sequence of iterated optimal stopping problems which is also introduced in Oksendal and Sulem (2004). We develop new results which allow us to reduce a given impulse control problem to a sequence of iterated optimal stopping problems even though the original process is affected by interventions.

In this thesis, the analysis of the dynamic response of a Linear Induction Motor as an electromechanical system is done, accounting for all the governing equations implied in the process which are used to develop the corresponding simulation models. Once this model is presented, a feedback control system is implemented in order to analyze the controlled response of the motor, considering the applications and conditions analogue to aircraft launcher systems. Also a comparison between the Linear and Rotary induction motors describing the differences, similarities and equivalences will be developed.