Synthesis and Analysis of Design Methods in Linear Repetitive, Iterative Learning and Model Predictive Control

Repetitive Control (RC) seeks to converge to zero tracking error of a feedback control system performing periodic command as time progresses, or to cancel the influence of a periodic disturbance as time progresses, by observing the error in the previous period. Iterative Learning Control (ILC) is similar, it aims to converge to zero tracking error of system repeatedly performing the same task, and also adjusting the command to the feedback controller each repetition based on the error in the previous repetition. Compared to the conventional feedback control design methods, RC and ILC improve the performance over repetitions, and both aiming at zero tracking error in the real world instead of in a mathematical model. Linear Model Predictive Control (LMPC) normally does not aim for zero tracking error following a desired trajectory, but aims to minimize a quadratic cost function to the prediction horizon, and then apply the first control action. Then repeat the process each time step. The usual quadratic cost is a trade-off function between tracking accuracy and control effort and hence is not asking for zero error. It is also not specialized to periodic command or periodic disturbance as RC is, but does require that one knows the future desired command up to the prediction horizon. The objective of this dissertation is to present various design schemes of improving the tracking performance in a control system based on ILC, RC and LMPC. The dissertation contains four major chapters. The first chapter studies the optimization of the design parameters, in particular as related to measurement noise, and the need of a cutoff filter when dealing with actuator limitations, robustness to model error. The results aim to guide the user in tuning the design parameters available when creating a repetitive control system. In the second chapter, we investigate how ILC laws can be converted for use in RC to improve performance. And robustification by adding control penalty in cost function is compared to use a frequency cutoff filter. The third chapter develops a method to create desired trajectories with a zero tracking interval without involving an unstable inverse solution. An easily implementable feedback version is created to optimize the same cost every time step from the current measured position. An ILC algorithm is also created to iteratively learn to give local zero error in the real world while using an imperfect model. This approach also gives a method to apply ILC to endpoint problem without specifying an arbitrary trajectory to follow to reach the endpoint. This creates a method for ILC to apply to such problems without asking for accurate tracking of a somewhat arbitrary trajectory to accomplish learning to reach the desired endpoint. The last chapter outlines a set of uses for a stable inverse in control applications, including Linear Model Predictive Control (LMPC), and LMPC applied to Repetitive Control (RC-LMPC), and a generalized form of a one-step ahead control. An important characteristic is that this approach has the property of converging to zero tracking error in a small number of time steps, which is finite time convergence instead of asymptotic convergence as time tends to infinity

2-local unstable homotopy groups of indecomposable A32\mathbf{A}_3^2 -complexes

In this paper, we calculate the 2-local unstable homotopy groups of indecomposable $\mathbf{A}_3^2$-complexes. The main technique used is analysing the homotopy property of $J(X,A)$, defined by B. Gray for a CW-pair $(X,A)$, which is homotopy equivalent to the homotopy fibre of the pinch map $X\cup CA\rightarrow \Sigma A$

A stirrer for magnetohydrodynamically controlled minute fluidic networks

Magnetohydrodynamics may potentially provide a convenient means for controlling fluid flow and stirring fluids in minute fluidic networks. The branches of such fluidic networks consist of conduits with rectangular cross sections. Each conduit has two individually controlled electrodes positioned along opposing walls and additional disk-shaped electrodes deposited in the conduit\u27s interior away from its sidewalls. The network is positioned in a uniform magnetic field. When one applies a potential difference between a disk-shaped electrode and two wall electrodes acting in tandem, circulatory motion is induced in the conduit. When the potential difference alternates periodically across two or more such configurations, complicated (chaotic) motions evolve. As the period of alternation increases, so does the complexity of the flow. We derive a two-dimensional, time-independent expression for the magnetohydrodynamic creeping flow around a centrally positioned disk-shaped electrode in the limit of zero radius. With the aid of this expression, the trajectories of passive tracers are computed as functions of the alternations protocol and the electrodes\u27 locations. The theoretical results are qualitatively compared with flow visualization experiments

A magneto-hydrodynamically controlled fluidic network

The paper describes fluidic networks consisting of individually controlled branches. The networks\u27 basic building blocks are conduits equipped with two electrodes positioned on opposing walls. The entire device is either subjected to an external uniform magnetic field or fabricated within a magnetic material. When a prescribed potential difference is applied across each electrode pair, it induces current in the liquid (assumed to be at least a weak electrolyte solution). Analogously with electric circuits, by judicious application of the potential differences at various branches, one can direct liquid flow in any desired way without a need for mechanical pumps or valves. Equipped with additional, internally located electrodes, the network branches double as stirrers capable of generating chaotic advection. The paper describes the basic building blocks for such a network, the operation of these branches as stirrers, a general linear graph-based theory for the analysis and optimal control of fluidic magneto-hydrodynamic networks, an example of a network fabricated with low temperature, co-fired ceramic tapes, and preliminary experimental observations that illustrate that the ideas described in this paper can, indeed, be implemented in practice