525.666.3VL - Linear System Theory

Expanded Course Description

Many physical and electrical systems in the world can be accurately modeled by linear differential equations. This is quit useful because a comprehensive theory exists to understand the ways and the limitations to altering the behavior of a linear dynamic system and thus achieve useful engineering objectives.  For example, an aircraft must maintain stable flight throughout a variety of external disturbances (e.g., extreme wind gusts) while allowing a pilot to securely handle a flight path.  The same ideas extend to unmanned aircraft or driverless ground vehicles.  This course will cover the basic theories and design techniques that enable an engineer to understand the behavior of a dynamic system and how that behavior can be altered in a desired way.  Practical examples will be introduced throughout the course.  These include aircraft control, space telescope pointing, and inertial navigation.

Course Structure

The overall course material is organized into four main topic areas.  The first is the description of linear dynamic systems using differential equations, input-output transfer functions, internal state-space descriptions, and block diagrams.  The second topic area covers the underlying linear algebraic concepts for analyzing and designing a linear dynamic system.  We next discuss the fundamental characteristics and properties of linear dynamic systems including the time solution, stability, controllability, and observability.  Finally, the last portion of the course covers state variable feedback and observer designs to achieve the desired behavior of a system.

Weekly lectures follow the course outline.

Course Topics

The topics are arranged weekly with pertinent text chapters listed in the schedule:

• Review of Laplace Transforms, Transfer Functions;  Review of Matrix Theory; State Equations, Transfer Functions, and Block Diagrams
• Linear Vector Spaces; Range and Null Spaces of a Matrix
• Orthogonal Projections; Solution of Linear Algebraic Equations; Eigenvalues and Eigenvectors, State Transformations    
• Change of Bases; Jordan Form of a Matrix; Solution of Linear Dynamic Systems; State Transition Matrix  
• Cayley-Hamilton Theorem; Controllability and Observability; Examples: Spacecraft Attitude Control, Inertial Navigation 
• Canonical Realizations, Controllability Grammian; Minimal Realizations; Canonical Decomposition
• System Internal and External Stability; State Variable Feedback 
• State Observers 
  
• Controller-Observer Compensator Design; Tracking Problems: Magnetically Levitated Train
• Introduction to Multivariable Systems; Discrete Time Systems
• Linear Quadratic Regulator
  

Course Goals

Students will obtain a thorough mathematical understanding of any system that can be described by linear differential equations. This understanding will further enable the learning of basic design techniques to both estimate the state of such systems and control their behaviors. 

Textbooks

 Not required, but a useful primary reference is  Linear State-Space Control Systems, Robert Williams and Douglas Lawrence  (John Wiley, 2007)

Other Materials & Online Resources

Other References:     Linear System Fundamentals, J. Gary Reid (McGraw-Hill)

                              Modern Control Theory, W. L. Brogan (Prentice-Hall)

                              Control System Design, Bernard Friedland

                              Linear Systems, Thomas Kailath (Prentice-Hall, 1980)

                              Linear Algebra and its Applications, G. Strang (Academic Press, 1980)

                             System Theory Linear and Design, C. T. Chen (Holt,Rinehart, and Winston, 1999) Oxford University Press   

Student Coursework Requirements

There will be 2 quizzes (30% of grade), a final exam (30%), and homework (40%). It should be noted that homework problems will constitute an essential part of the coursework to understand principles and techniques.  Homework will be distributed weekly and is due the following Monday by 4:00 pm.  Because solutions will be discussed that same evening, late submissions cannot be accepted in fairness to the other students.  Under extenuating circumstances, accommodations can be made.  Homework can be handed in late, but with a grading penalty.  Typically, each assignment will consist of 5 problems with each problem graded as 0, 0.25, 0.5, 0.75 or 1, depending on the degree of understanding shown.

The quizzes will be in class.  For those attending remotely, there will be a fixed simultaneous time window to download the exam and return the answers (scan and upload) by a set time. 

All Lectures, Lecture Videos, Assignments, and Solutions will be posted on Canvas.

Grading Policy

The quiz, homework grades, and the final exam percentages will be totaled per the stated weightings to find a total percentage.  As an illustration, the total percentage might translate to a letter grade according to:

100-97= A+;  96-93= A;  92-90= A−;  89-87= B+;  86-83= B;  82-80= B−;  79-77= C+; 76-73= C; 72-70= C; −69-67= D+;  66-63= D; <63= F

The actual boundaries may be modified based on instructor's judgement of overall class performance and an individual's grade pattern (e.g., presence of an "outlier")

Academic Policies

Deadlines for Adding, Dropping and Withdrawing from Courses

Students may add a course up to one week after the start of the term for that particular course. Students may drop courses according to the drop deadlines outlined in the EP academic calendar (https://ep.jhu.edu/student-services/academic-calendar/). Between the 6th week of the class and prior to the final withdrawal deadline, a student may withdraw from a course with a W on their academic record. A record of the course will remain on the academic record with a W appearing in the grade column to indicate that the student registered and withdrew from the course.

Academic Misconduct Policy

All students are required to read, know, and comply with the Johns Hopkins University Krieger School of Arts and Sciences (KSAS) / Whiting School of Engineering (WSE) Procedures for Handling Allegations of Misconduct by Full-Time and Part-Time Graduate Students.

This policy prohibits academic misconduct, including but not limited to the following: cheating or facilitating cheating; plagiarism; reuse of assignments; unauthorized collaboration; alteration of graded assignments; and unfair competition. Course materials (old assignments, texts, or examinations, etc.) should not be shared unless authorized by the course instructor. Any questions related to this policy should be directed to EP’s academic integrity officer at ep-academic-integrity@jhu.edu.

Students with Disabilities - Accommodations and Accessibility

Johns Hopkins University values diversity and inclusion. We are committed to providing welcoming, equitable, and accessible educational experiences for all students. Students with disabilities (including those with psychological conditions, medical conditions and temporary disabilities) can request accommodations for this course by providing an Accommodation Letter issued by Student Disability Services (SDS). Please request accommodations for this course as early as possible to provide time for effective communication and arrangements.

For further information or to start the process of requesting accommodations, please contact Student Disability Services at Engineering for Professionals, ep-disability-svcs@jhu.edu.

Student Conduct Code

The fundamental purpose of the JHU regulation of student conduct is to promote and to protect the health, safety, welfare, property, and rights of all members of the University community as well as to promote the orderly operation of the University and to safeguard its property and facilities. As members of the University community, students accept certain responsibilities which support the educational mission and create an environment in which all students are afforded the same opportunity to succeed academically. 

For a full description of the code please visit the following website: https://studentaffairs.jhu.edu/policies-guidelines/student-code/

Classroom Climate

JHU is committed to creating a classroom environment that values the diversity of experiences and perspectives that all students bring. Everyone has the right to be treated with dignity and respect. Fostering an inclusive climate is important. Research and experience show that students who interact with peers who are different from themselves learn new things and experience tangible educational outcomes. At no time in this learning process should someone be singled out or treated unequally on the basis of any seen or unseen part of their identity. 
 
If you have concerns in this course about harassment, discrimination, or any unequal treatment, or if you seek accommodations or resources, please reach out to the course instructor directly. Reporting will never impact your course grade. You may also share concerns with your program chair, the Assistant Dean for Diversity and Inclusion, or the Office of Institutional Equity. In handling reports, people will protect your privacy as much as possible, but faculty and staff are required to officially report information for some cases (e.g. sexual harassment).

Course Auditing

When a student enrolls in an EP course with “audit” status, the student must reach an understanding with the instructor as to what is required to earn the “audit.” If the student does not meet those expectations, the instructor must notify the EP Registration Team [EP-Registration@exchange.johnshopkins.edu] in order for the student to be retroactively dropped or withdrawn from the course (depending on when the "audit" was requested and in accordance with EP registration deadlines). All lecture content will remain accessible to auditing students, but access to all other course material is left to the discretion of the instructor.