An Introduction to Control Theory From Classical to Quantum Applications Course Lecture Notes

Control theory is a vital component of all modern technology because it allows the construction of high performance devices despite large uncertainty in the individual components used to build them. This course provides a basic introduction to modern control theory in a series of five lectures, each intended to be roughly three hours long. It focusses on the development of fundamental concepts, such as stability, performance and robustness, that lie at the heart of the subject. However, it also addresses practical issues in controller design and presents techniques for constructing and improving control systems typically found in an experimental physics setting. In the first part of the course (Lectures 1-3) we develop basic definitions in classical dynamical systems and focus on linear feedback control. Experience has shown that a vast majority of physical systems, even those that are nonlinear, can be well controlled using a feedback theory of this type. We explore quantitative descriptions of stability, performance and robustness and use them to develop design criteria for engineering feedback controllers. In Lecture 3, we work through a full example of this loop-shaping design procedure. The second part of the course (Lectures 4-5) makes a transition to quantum systems. As modern devices grow ever smaller, and due to the realization that quantum mechanics opens avenues for computation and communication that exceed the limits of classical information theories, there has been substantial interest in extending the field of feedback control to include quantum dynamical systems. This process is already being realized in both theoretical and experimental quantum optics for precision measurement and fundamental tests of quantum measurement theories. Forefront research in microand nano-scale physics has produced mechanical systems that are rapidly approaching the quantum regime. We will look at the formulation of quantum feedback control theory for continuously observed open quantum systems in a manner that highlights both the similarities and differences between classical and quantum control theory. The final lecture will involve a discussion of special topics in the field and is meant to provide a casual overview of current experiments in quantum control.