Teaching > ECE311F
ECE311F - Introduction to Linear Control Systems (Fall 2021)
Calendar Description
3/1.50m/1m/0.50
III,IV-AECPEBASC, III,IV-AEELEBASC
An introduction to dynamic systems and their control. Differential equation models of mechanical, electrical, and electromechanical systems. State variable form. Linearization of nonlinear models and transfer functions. Use of Laplace transform to solve ordinary differential equations. Conversion of models from state variable form to transfer function representation and vice versa. Block diagrams and their manipulation. Time response: transient analysis and performance measures. Properties of feedback control systems. Steady state tracking:the notion of system type. The concept of stability of feedback systems, Routh-Hurwitz stability criterion. Frequency response and stability in the frequency domain. Root locus. Bode and Nyquist plots and their use in feedback control design.
Prerequisite: MAT290H1, MAT291H1, ECE216H1
Graduate Attributes
- Knowledge base:
- Demonstrate competence in mathematics and modeling
- Demonstrate competence in natural sciences and engineering fundamentals
- Problem analysis:
- Demonstrate the ability to identify and characterize an engineering problem
- Demonstrate the ability to formulate and interpret a model
- Use of Engineering tools:
- Demonstrate ability to use discipline specific techniques, resources and engineering tools
Reference: UofT Engineering Graduate Attributes Poster
Learning Objectives
Imagine a humanoid robot that walks, climbs stairs, sits down and stands up; a quadrotor helicopter that hovers autonomously at a fixed distance from the ground; a self-driving car that keeps a lane while maintaining a desired cruise speed. These are all examples of control systems, machines endowed with sensors and actuators, and running an algorithm, the controller, that reads sensor data and decides, in real-time, how to drive the actuators so as to achieve a desired objective. The decisions of the controller affect the behaviour of the machine, producing new sensor data that in turn affect future decisions by the controller. This unending decision making loop is called a feedback control loop.
This course is an introduction to feedback control loops such as the ones described above, and it presents the indispensable tools required to design controllers, algorithms forming the brain of any device that is to function autonomously, be it a robot, a quadrotor helicopter, or a self-driving car. The course offers a window into the fascinating field of Control Theory, the discipline that aims to develop universal tools for solving problems of the kind just described.
In this course, we focus on linear time-invariant systems, and in particular on their transfer function representation, leveraging tools that you have acquired in ECE216 and MAT290. The course material is divided into four chapters.
- LTI Control Systems: Models and Basic Properties: We will present a number of examples of control systems and motivated by these, we will present different mathematical representations of control systems. We will then introduce linear time-invariant (LTI) control systems, and focus on their transfer function representation using Laplace transforms. We will learn how to quickly predict the qualitative properties of the output of an LTI control system based on the poles of its transfer function. We will learn how to represent interconnections of transfer functions using block diagrams.
- Stability: The most basic properties one seeks in a control system is stability. We will present two types of stability, internal and input-output. We will learn how to easily check whether a control system is input-output stable through the so-called Routh stability test.
- Tracking with Disturbance Rejection: Armed with the concept of stability, we will discuss the "basic problem" of linear control theory, tracking with disturbance rejection. Roughly speaking, the problem is to make the output of the control system asymptotically converge to a desired "reference" signal while rejecting a partially unknown disturbance. We will see that as long as the controller incorporates an internal model of the reference and disturbance signals, then solving the problem amounts to designing a controller rendering the control system input-output stable.
- Stabilization by frequency domain methods: The outcome of the previous chapter is that in order to solve the "basic problem" one needs to know how to design controllers that render the control system input-output stable. In this chapter, we will do just that, relying on the so-called Nyquist stability criterion. This criterion leads to a control design methodology based on the frequency response of the control system and its graphical representation using Bode diagrams, all notions that you have already encountered in ECE216. We will present a number of canonical control structures: PID and lead-lag controllers.
Instructors
M. Maggiore (LEC0101)
Office: GB344
Email address: maggiore (at)
ece.utoronto.ca
L. Pavel (LEC0102)
Office: GB343A
Email address: pavel (at) ece.utoronto.ca
Teaching Assistants
| Dian Gadjov | dian.gadjov@mail.utoronto.ca |
| Mohamed Hafez | mohamed.hafez@mail.utoronto.ca |
| Adan Moran-MacDonald | adan.moran@mail.utoronto.ca |
| Rein Otsason | rein.otsason@mail.utoronto.ca |
| Andrew Romano | andrew.romano@mail.utoronto.ca |
| Gianluca Villani | gianluca.villani@mail.utoronto.ca |
| Emily Vukovich | emily.vukovich@mail.utoronto.ca |
Lectures
| M. Maggiore | LEC0101 |
|---|---|
| Tue 3-4PM | BA1210 |
| Thu 12-1PM | BA1210 |
| Fri 3-4PM | GB220 |
| L. Pavel | LEC0102 |
|---|---|
| Mon 11-12 | BA1240 |
| Tue 9-10 | MP137 |
| Thu 9-10 | MP137 |
Composition of Final Mark
| Labs | 20% |
| Homework Assignments | 10% |
| Quiz | 10% |
| Midterm Exam | 20% |
| Final Exam | 40% |
Textbook
B.A. Francis, Classical Control, available on Quercus.
Additional Reference Text
K.J. Astrom and R.M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, online edition available here.
Detailed Course Outline
- Introduction to feedback, block diagrams, and state models
- Linearization of a nonlinear system at an equilibrium
- Review of Laplace transforms and their use in solving linear constant coefficient differential equations
- Transfer functions and block diagram manipulations
- The time response of a linear time-invariant system
- Stability
- The concept of feedback. Proportional control design
- Asymptotic tracking: the internal model principle
- The Nyquist stability criterion
- Frequency response and Bode plots
- Lead, lag, and PID controller design
Calendar of deliverables
| Deliverable | Due date |
|---|---|
| Assignment 1 | Sep 29 |
| Assignment 1: Self-assessment | Oct 5 |
| Lab 1 | Oct 6 |
| Assignment 2 | Oct 15 |
| Assignment 2: Self-assessment | Oct 18 |
| Quiz | Oct 19 |
| Lab 2 | Oct 26 |
| Assignment 3 | Nov 2 |
| Assignment 3: Self-assessment | Nov 5 |
| Lab 3 | Nov 19 |
| Midterm test | Nov 23 |
| Assignment 4 | Nov 25 |
| Assignment 4: Self-assessment | Nov 30 |
| Assignment 5 | Dec 3 |
| Lab 4 | Dec 6 |
| Assignment 5: Self-assessment | Dec 8 |
Late Submission Policy
We do not accept late online submissions, under any circumstance. We do not accept submissions via email. This policy is strictly enforced for labs and assignments. A late submission will receive a mark of 0.
If you deem it unavoidable to submit a deliverable after the deadline, you need to contact the instructor before the deadline of the deliverable, explain the circumstances surrounding the expected delay, and check whether or not the instructor gives you permission to submit late. In the absence of such an advance permission, the policy above applies.
Tests
There will be a quiz, a midterm test, and a final exam. The quiz will be 1 hour long, the midterm test will be 2 hours long, and the final exam will be 2.5 hours long.
| Day and Time | Room | |
|---|---|---|
| Quiz | Oct 19, 6-7:10PM | EX200 |
| Midterm test | Nov 23, 6-8PM | EX200 |
Tutorials
| Section | When | Where | Start on |
|---|---|---|---|
| TUT0101 | Fri 12-1PM | GB303 | Sep 17 |
| TUT0102 | Wed 2-3PM | BA1220 | Sep 22 |
Homework Assignments
There will be five homework assignments posted on Quercus. The marking will be based on two components: submission (1 point) and self-evaluation (1 point). For the submission component, a full and clearly legible solution will be given full marks (1 out of 1) independently of its correctness. Poorly written or incomplete assignments will not be given credit. For the self-evaluation component, after the assignment is due we will post the solutions. Using our solutions, you will resubmit a version of your assignment containing highlights in red colour of any mistakes and omissions. You will receive full credit for a resubmission containing adequate and clearly legible commentary. Homework assignments are subject to the late submission policy.
Laboratories
You will perform four labs in this course, all of them Matlab-based. The lab documents are posted on Quercus.
Labs are performed in groups of three students. You will join a group by using the group self sign-up feature found in Quercus->People. Members of each group get identical marks, unless special circumstances occur. The group members do not need to be in the same practical section.
Each lab group will work remotely. Each lab has three office hour slots in BA3114 that you can attend to ask questions and clarifications. Lab TAs will not be able to answer questions sent via email so the lab office hours will be your only opportunity to seek help and clarification.
| Office hour slot 1 | Office hour slot 2 | Office hour slot 3 | Lab due date | |
|---|---|---|---|---|
| Lab 1 | Fri Oct 1, 1-3PM | Mon Oct 4, 10-12PM | Tue Oct 5, 1-3PM | Wed Oct 6 |
| Lab 2 | Fri Oct 22, 1-3PM | Mon Oct 25, 10-12PM | Tue Oct 26, 1-3PM | Tue Oct 26 |
| Lab 3 | Fri Nov 19, 1-3PM | Mon Nov 15, 10-12PM | Tue Nov 16, 1-3PM | Fri Nov 19 |
| Lab 4 | Fri Dec 3, 1-3PM | Mon Dec 6, 10-12PM | Tue Nov 30, 1-3PM | Mon Dec 6 |
Your lab grade will be based on three components:
- The Matlab and Simulink work by the group.
- The lab report.
- The presentation of your work in a five-minute video.
Labs are subject to the late submission policy.
Lab 1 shows you how to define linear control systems structures in Matlab, using either state space or transfer function representations, how to transition from one of these representations to the other, and how to simulate the response of an LTI system. The problem you will investigate in the lab is speed control of a permanent magnet DC motor.
Lab 2 shows you how to define a nonlinear control system in Simulink, and how to linearize it at an equilibrium. The example in question is that of a basic magnetic levitation system in which an electromagnet is used to magnetically levitate a steel ball. You will design a lead controller to levitate the ball.
Lab 3 introduces you to P and PI control design and the Internal Model Principle. The problem here is to design a cruise control system for a car, which regulates the speed of the car to a desired value irrespective of the unknown slope of the road. You will need to meet certain transient performance specifications by choosing desired locations for the poles of the closed-loop system.
Lab 4 concerns the design of a servomotor based on a permanent magnet DC (PMDC) motor. The problem is to design a controller for a PMDC in order to regulate the shaft angle to a desired value. You will have tight control specifications to meet. You will also be exposed here to the concept of integrator antiwindup, a clever mechanism to avoid actuator saturation.