# Academics  /  Courses  /  DescriptionsELEC_ENG 410: System Theory

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### Prerequisites

ELEC_ENG 360 or equivalent

### 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, 4th Edition, 2013 or International 4th edition, 2013. The international edition will probably have to be ordered from the UK. The two editions are not the same but that will be taken into account. You may encounter large price differences.

REFERENCE TEXTS: None

COURSE DIRECTOR: Prof. 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: ELEC_ENG 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.