Geometric Control Theory

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This course is the continuation course of Mathematical Systems Theory. More information can be found on the Canvas page for the course.

Linear geometric control theory was initiated in the beginning of the 1970's. A good summary of the subject is the book by Wonham.

The term ``geometric'' suggests several things. First it suggests that the setting is linear state space and the mathematics behind is primarily linear algebra (with a geometric flavor). Secondly it suggests that the underlying methodology is geometric. It treats many important system concepts, for example controllability, as geometric properties of the state space or its subspaces. These are the properties that are preserved under coordinate changes, for example, the so-called invariant or controlled invariant subspaces. On the other hand, we know that things like distance and shape do depend on the coordinate system one chooses. Using these concepts the geometric approach captures the essence of many analysis and synthesis problems and treats them in a coordinate-free fashion. By characterizing the solvability of a control problem as a verifiable property of some constructible subspace, calculation of the control law becomes much easier. In many cases, the geometric approach can convert what is usually a difficult nonlinear problem into a straight-forward linear one.

As we can already see, "invariance" is a very important concept in the geometric control theory. It is not coincidence that invariance is also very important in machine learning since it preserves the object's identity, category and so forth across changes in the specifics of the input. In this respect, geometric methods are also important in machine learning.

The linear geometric control theory was extended to nonlinear systems in the 1970's and 1980's (see the book by Isidori). The underlying fundamental concepts are almost the same, but the mathematics is different. For nonlinear systems the tools from differential geometry are primarily used.

The course compendium is organized as follows.

Chapter 1 is introduction; In Chapter 2, invariant and controlled invariant subspaces will be discussed; In Chapter 3, the disturbance decoupling problem will be introduced; In Chapter 4, we will introduce transmission zeros and their geometric interpretations; In Chapter 5, non-interacting control and tracking will be studied as applications of the zero dynamics normal form; In Chapter 6, we will discuss some input-output behaviors from a geometric point of view; In Chapter 7, we will discuss the output regulator problem in some detail; In Chapter 8, we will extend some of the central concepts in the geometric control to nonlinear systems. Finally, in Chapter 9 some applications to mobile robots will be given.