||Mathematical systems theory is concerned with problems related to dynamic phenomena in interaction with their environment. These problems include:
* Modeling. Obtaining a mathematical model that reflects the main
features. A mathematical model may be represented by difference or
differential equations, but also by inequalities, algebraic equations,
and logical constraints.
* Analysis and simulation of the mathematical model.
* Prediction and estimation.
* Control. By choosing inputs or, more general, by imposing additional
constraints on some of the variables, the system may be influenced so as to obtain certain desired behavior. Feedback is an important example of control.
The main objects of study in this course are systems modeled by linear time-invariant differential equations. We start with a treatment of the theory of algebraic representation of dynamical
systems using polynomial matrices. The main tool is the Euclidean
algorithm applied to matrices of real polynomials. The main result is a
complete characterization of all representations of a given system.
Several other representation are introduced along with their relations.
Important examples of such representation are input-output
representations that reveal that some variables may be unrestricted by the equations, and state space representations that visualize the
separation of past and future, also referred to as the Markov property.
Controllability and observability are important system theoretic
concepts. A controllable system has the property that a desired future behavior can always be obtained, independent of the past behavior,
provided that this future behavior is compatible with the laws of the
system. Observability means that the complete behavior may be
reconstructed from incomplete observations. The theory of
controllability and observability forms one of the highlights of the
course. Stability can be an important and desirable property of a
system. Stabilization by static or dynamic feedback is one of the key
features of Systems and Control. In the pole placement theorem linear algebraic methods and the notion of controllability are used in their full strength. The theorem, loosely speaking, says that in a
controllable system the dynamic behavior can be changed as desired, in terms of characteristic values, by using appropriate feedback. It forms one of the most elegant results of the course and indeed of the field of Systems and Control.
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