The purpose of this course is to provide an understanding of the theory of linear systems from the state space perspective, with specific application toward feedback control design. Although mathematics (particularly linear algebra) is the language by which the theory is described, this is not mathematics course. The theorem/proof format is avoided in favor of an exposition of useful "truths" and a demonstration of the underlying reasons. The geometry and insight behind matrix algebra, in particular, is stressed. However, expect to learn a little math in the process.

The understanding sought in this course is a foundation for further graduate work in various fields, particularly nonlinear dynamical systems, data analysis, advanced control systems, etc. It introduces standard viewpoints, methods, and terminology used in the applied and research literature. It also provides the basis for understanding how many computational analysis and design tools work.

• Develop some expertise with the state space modeling/analysis/design approach, learning to see dynamical systems in a new way with new concepts, vocabulary, tools, and insights.
• See linear algebra in a new light, where matrices are representations of linear operators, and these operators have simple geometry and corresponding insights.
• Glimpse how optimization can be used to design control systems "automatically"
• Understand how applications of this theory can be limited by inaccuracy in system models.

After taking this course, you should be able to:

1. construct state space models from differential equations and transfer functions.
2. test a set for vector space properties, apply the concepts of linear independence, subspace, span, dimension, basis.
3. test a mapping from one vector space to another for linearity, apply change of basis, spectrally decompose a generic linear mapping using eigenspaces.
4. use concepts of column space and null space to characterize solutions of linear equations, apply this to the solution of state space equations via Laplace transforms, show how eigenvalues of the state matrix and related to transfer function poles.
5. use a modal basis to derive the general solution to state space equations and to prove the Cayley-Hamilton theorem.
6. characterize the Lyapunov stability properties of state space systems, use Lyapunov functions to design control systems with prescribed settling time properties.
7. understand the tests for complete controllability and observability, apply them to find controllable and unobservable subspaces.
8. design observers to reconstruct internal states.
9. design state feedback controllers to achieve prescribed closed loop poles.
10. use linear-quadratic optimization to design control systems.
11. understand limitations of pole placement/optimization theory due to unmodeled dynamics.
12. use MATLAB as an aid in solving numerical problems associated with the above concepts, simulating system responses, and computing state feedback/optimal controllers.