**Title:** LINEAR MULTIVARIABLE SYSTEMS

**Credits:** 3

**Catalog Description: **EE 454 Linear Multivariable Systems (3+0+0) 3

Fundamentals of polynomial matrix theory. Finite and infinite pole/zero structure of transfer matrices. Realization theory: minimality and minimal realizations of transfer matrices. Linear state feedback design, linear quadratic regulator problem, design of observer-based compensators for multi-input, multi-output linear systems.

**Prerequisite:** EE 453

**Coordinator:** Kadri Özçaldıran, Professor of Electrical Engineering

**Goals:** This course aims to expose the senior year students to analysis and design of linear multivariable systems.

**Learning Objectives:**

At the end of this course, students will be able to:

CLO 1- Construct an irreducible (right or left) matrix fraction description of a given proper transfer function matrix and use it to find a realization of it.

CLO 2 - Construct the Smith-McMillan Form of a given transfer function matrix H(s), and compute poles and transmission zeroes of H(s).

CLO 3 - Construct the Brunovsky Form of a given MI realization and use it to design a state feedback controller.

CLO 4 - Solve the Linear Quadratic Regulator Problem for linear, time-invariant systems in continuous time.

**Textbook:** T. Kailath, Linear Systems, Prentice-Hall, 2000.

**Reference Texts:**

1. W.L. Brogan, Modern Control Theory, 3rd. edition, Prentice-Hall,1991.

2. W.J. Rugh, Linear System Theory, 2nd. Edition, Prentice-Hall, 1996.

3. P.J. Antsaklis and A.N. Michel, Linear Systems, Mc Graw Hill, 1997.

4. C.-T. Chen, Linear System Theory and Design, Holt-Saunders, 1984.

**Prerequisites by Topic:**

1. Linear algebra

2. Classical control theory

3. Laplace Transform

4. Linear System Theory for SISO systems

**Topics:**

1. Polynomial matrix fraction descriptions. Right and left coprimeness. Irreducibility. Column (row) reducedness. Hermite forms. Smith form. (5 weeks)

2. Multivariable realization theory. (2 weeks)

3. Smith-McMillan form. Poles and zeroes of multivariable linear systems. (1 week)

4. Feedback design for MIMO systems. Brunovsky Canonical Form. (2 weeks)

5. Introduction to optimal control. LQR Problem for continuous-time systems. (3 weeks)

**Course Structure:** The class meets for three lectures a week, each consisting of two 50-minute sessions. A number of homework sets will be handed out as the course progresses. Homeworks will not be collected and graded. They will be discussed in detail collectively in class.

**Computer Resources:** Students are encouraged to use MATLAB to solve their homework problems.

**Laboratory Resources**: None.

**Grading:**

Letter grade assignment will be a weighted average of the following:

1. In-class participation during lectures and during homework sessions. (50%)

2. A mid-term exam (20%).

3. A final exam (30%).

**Outcome Coverage:**

Outcome (a): Apply math, science and engineering knowledge.

This course aims to generalize the results of EE 453, which basically deals with SISO systems, to MIMO systems, and in doing so, utilizes results from not only linear algebra and ordinary differential equations but also from the theory of rings.

**Relation Between CLOs and Outcomes:**

All four CLOs are meant to support and strengthen Outcome (a).

**Relation Between CLOs and Program Objectives:**

All four CLOs of this course serve the first two educational objectives of the program that state that graduates of the program will

(i) have a strong background in basic sciences, mathematics and engineering to be successful in their graduate studies;

(ii) have broad skills and solid technical background to be successful in their professional careers.

All four CLOs of EE 454 guarantee that program objective (i) is served adequately in the sense that they strengthen mathematical background of the student, and they prepare the student for graduate work in system theory and controls. It is next to impossible to envision a doctoral candidate in systems and controls who has not mastered this contents of EE 454 completely.

Alson, all four CLOs, but in particular CLO 3 and CLO 4, provide the students with convenient, important and useful tools that they will use in their professional careers as control engineers, thus serving program objective (ii).

**Prepared By:** Kadri Özçaldıran