## EECS 410 - System Theory

**CATALOG DESCRIPTION: **Unified treatment of continuous and discrete time systems from a state-variable viewpoint; emphasis on linear systems. Concept of state, writing and solving state equations, controllability and observability, transform techniques (Fourier, Laplace, Z), stability, and Lyapunov's method.

REQUIRED TEXTS: Chen, C. T., Linear System Theory and Design, Oxford, 4^{th} Edition, 2013

**REFERENCE TEXTS**: None.

**COURSE DIRECTOR:** Arthur Butz

**COURSE GOALS**: Describe linear dynamic systems in terms of state variables and vector-matrix differential equations (continuous time) or difference equations (discrete time). The topics will include concept of linear space and linear operators, matrix algebra, eigenvalues and generalized eigenvectors, matrix functions, representation and solution of state-variable dynamic equations, controllability and observability of linear dynamic systems, and stability considerations

**PREREQUISITES BY COURSES**: EECS 360 or equivalent

**PREREQUISITES BY TOPIC**:

- Linear algebra
- Transfer functions of linear time-invariant systems, poles and zeros, Laplace and Z transforms

**DETAILED COURSE TOPICS:**

- Introduction to state-space systems, differences between state-space and input-output models of systems
- Linear state-space models, small-signal linearization, similarity transformations
- State transition matrices for continuous- and discrete-time linear systems, solutions to state equations
- Linear time-invariant systems, matrix exponential
- Computation of matrix exponential via Jordan form and Laplace/Z transforms, functions of a square matrix, Cayley-Hamilton theorem
- Stability
- Controllability, observability, and reachability of linear systems
- Realizability, minimal realizations
- Canonical realizations, Kalman decomposition

**COMPUTER USAGE: **at the discretion of the instructor.

**LABORATORY PROJECTS: **None.

**GRADES: **Weights on homework, exams, etc. are at the discretion of the instructor.

**COURSE OBJECTIVES: **When a student completes this course, s/he should be able to:

- analyze linear state-space systems in continuous- and discrete time (controllability, observability, reachability, stability.
- solve linear time-invariant state equations in continuous- and discrete time.
- obtain state-space realizations of transfer functions and likewise derive transfer functions from state-space descriptions.