ECE60200 - Lumped System Theory**Fall 2016** **Days/Time:** TTh / TBA

**Credit Hours:** 3

**Learning Objective:**1. Promote in-depth understanding of basic calculations, applications, & theoretical concepts of state variable approach to system theory from basic vocabulary & examples to general solution of linear & time varying state model, to critical material on controllability, observability, & stability.

2. In order to achieve "1", the student must

(i) understand the basic vocabulary of system theory & state variable model formulation

(ii) apply basic principles & techniques to simple and complex modeling & analysis problems

(iii) theoretically analyze complex state variable problems, re: (a) computing solutions, (b) controllability, (c) observability, and (d) stability

(iv) critically evaluate procedure used to obtain accurate solutions.

3. Foster developmental thinking & pressured creativity in problem solving, i.e. by giving someone a fish, one provides a single meal; by teaching someone to fish, one provides food for a lifetime.

**Description:**Set forth the basic theory, numerical calculations, and applications of linear time invariant and time-varying state model formulations of continuous and discrete time systems. This includes some transfer function matrix material when defined, along with some nonlinear material, e.g., linearization of a nonlinear state model. Develop analytic solutions of state models including the notion of the state transition matrix and the complete solution to the linear state model. In the time invariant case this includes an extensive eigenvalue-eigenvector discussion of the matrix exponential. Finally the course develops the analytic and numerical content of the critical concepts of controllability, observability, and stability. Students should be able to apply the material to determine the controllability, observability and stability of complex systems in state variable form.

**Topics Covered:**1. Basic Concepts, Vocabulary, Notation, and Examples

2. Nonlinear and Linearized/Linear State Model Basics

3. Solution of the Linear Time Invariant State Model: Motivation and Basics

4. Equivalent State models, canonical forms, an intro to impulse response and transfer functions, state model identification, and pole placement by state feedback

5. Computing and Setting Up Initial States: A Naive Introduction to observability and controllability using the singular value decomposition of a matrix as a way to understand the solution of the equations.

6. Basics of the linear (constant coefficient) discrete time state model

7. Existence and Uniqueness of state trajectories

8. The State Transition Matrix, Fundamental Matrices, and the time varying state model

9. An eigenvalue-eigenvector method for computing the matrix exponential and a decomposition into matrix directed modes

10. The Jordan form

11. The impulse response and transfer function matrices

12. Controllability state control of the time invariant state model with relationship to the time varying and switched system cases.

13. Observability and observer design for linear time invariant systems.

14. Stability of lumped time invariant systems

**Prerequisites:**1. Basic knowledge of signals and systems: Laplace transforms, Z-transforms, impulse response, transfer functions, convolution, difference equations, differential equations, etc.

2. A minimal knowledge of basic circuit theory.

3. Firm knowledge of linear algebra:

(i) Matrices and Vectors

(ii) Eigenvalues and Eigenvectors

(iii) Rank, column space, determinants, matrix inversion, etc.

4. Some knowledge of complex variables including partial fraction expansions.

5. A willingness to stretch your current knowledge and thinking patterns.

**Applied/Theory:** 25/75

**Homework:**Assignments handed out on weekly basis, generally containing 2 types of problems: required & suggested. It is assumed that required problems will be completed by due date. All encouraged to at least attempt suggested homework problems. Homework will NOT be collected but is covered on exams.

**Projects:**None required; however, please note: a project may be given out as extra credit as determined later in the semester.

**Exams:**Four midterms and one final exam. Makeup exams NOT given and missed exam counts zero. All exams are closed book and no calculator.

**Textbooks:**No required textbook. **Updated 5/03/2012* Recommended Text: "Linear System Theory and Design" by Chi-Tsong Chen, Oxford University Press, 1999. ISBN: 0195117778. Lecture notes will be mailed to students or sent as PDF documents via e-mail.

**Computer Requirements:**ProEd minimum computer requirements; student edition of MATLAB or its equivalent (e.g., Mathematica).

**ProEd Minimum Requirements:** view**Tuition & Fees:** view**Other Requirements:**None.