||Advanced System Theory
||Dr.ing. D. Jeltsema, Dr. J.W. van der Woude
|Contact Hours / Week x/x/x/x:
||Exam by appointment
In part I of this course the connection of linear system theory and convex optimization is illustrated. One of the key ingredients are the so-called Linear Matrix Inequalities, LMI’s for short. LMI’s can be treated efficiently by means of semi-definite programming techniques coming from convex optimization.
It turns out that many properties of linear systems, like stability, controllability, observability, etc., can be formulated in terms of LMI’s. Also the design of controllers satisfying stability and other constraints can be done efficiently using semi-definite programming and LMI’s. The first part of part I starts by recalling basic knowledge from linear system theory and placing it in the frame work of LMI’s.
A second topic in part I will be LQ optimal control and the introduction of dissipativity. Both topics are of crucial importance for system theory. LQ optimal control has a long and rich history, but is still important and applicable. Dissipativity also has a long history, but its applicability has increased in recent years by the event of new efficient algorithms to solve semi-definite programming problems.
The last topic in part I are system norms and the design of a controller such that the combined system behaves in a desired way specified in terms of its norm. To that end, the H∞ - and the H2 norm will be introduced. Also methods will be treated which it can be investigated whether a certain desired norm can be achieved, and how this then actually can be done by means of state or output feedback.
The first fundamental topic to be treated in part II of the course is concerned with the study of the system"s internal behavior via Lyapunov stability theory. The extension of Lyapunov stability theory to systems with inputs and outputs will be accomplished by the introduction of the concept of dissipative systems. The two main examples of dissipative systems are passive systems and nonlinear control systems having finite input-output L2-induced norm. Important results, such as the small-gain theorem, are highlighted and implications towards the stability analysis of large-scale physical systems, as well as to the robustness of stability with respect to unmodeled dynamics are discussed.
Another main topic concerns the extension of the controllability and observability concepts to nonlinear control systems. The key ingredients to analyze controllability of a nonlinear system are the so-called Lie brackets of the associated system vector fields. Observability can be analyzed by considering the (repeated) Lie derivatives of the output mapping with respect to the system vector fields. The necessary mathematical preliminaries are introduced during the lectures.
In the last part of the course, the problem of transforming a nonlinear control system by feedback transformations and the choice of state space coordinates into a linear control system is discussed. It turns out that for controllable systems an elegant ‘if and only if’ condition can be given, stated in terms of the involutivity of certain Lie bracket expressions of the system vector fields. Applications with respect to control problems such as tracking of desired output trajectories will be provided.
||Advance system theory deals with the extensions in /of linear system theory.
After the course the student will be familiar with the following topics.
• Convex optimization and linear matrix inequalities
• LQ optimal control and dissipativity
• Controller design and the H-infinity norm/H-2 norm
• Passive systems/ nonlinear control systems and the small-gain theorem
• Controllability/observability for nonlinear systems and Lie brackets
• Feedback linearization and involution