Introductory Control Systems

Below are a set of links to notes, exercises, and equation sheets on elementary control systems. The topics covered are those typically covered in a first course in control systems in engineering curricula. Selected examples are included in the notes. Some schools also cover state-space methods. An introduction to state-space methods is provided at the end of the Notes section.

Reference: Dorf & Bishop, Modern Control Systems, 12th edition, Prentice-Hall, Inc, 2011

ME360Figure Roll Angle Control
Aircraft Roll Angle Control

Trademarks: MATLAB is a registered trademark of The MathWorks, Inc. The MathWorks does not warrant the accuracy of the examples given in these notes.


Knowledge Map

Map of Areas Related to Control Systems

Introduction to Control Systems

Control Systems Terminology

Introduction to Mathematical Modeling of Dynamic Systems

  1. Mathematical Models
  2. Linearity Principles
  3. Linearization of Algebraic Models
  4. Linearization of Equations of Motion of Pendulum and Accelerating Pendulum
  5. Model for a Hydraulic Positioning System

Short Review of Laplace Transforms

  1. Using Laplace Transforms to Solve Linear, Ordinary Differential Equations
  2. Laplace Transform Table
  3. Laplace Transform Examples
  4. Introduction to Partial Fraction Expansions
  5. Examples using Partial Fraction Expansions

Introduction to Transfer Functions and Block Diagrams

  1. Transfer Functions
  2. Cramer’s Rule for Solving a System of Linear Equations
  3. Block Diagrams and Transfer Functions
  4. Block Diagrams & Transfer Functions Example
  5. Block Diagram Reduction Transformations
  6. Block Diagram Reduction Examples
  7. Armature-Controlled DC Motor Transfer Function

Proportional Control of the Position of a Spring-Mass-Damper System

  1. Position Control of Spring-Mass-Damper using Proportional Control
  2. Plotting a Step Response Using MATLAB

General Characteristics of Systems with Feedback

  1. Characteristics of Open-Loop and Closed-Loop Systems
  2. Design Problem – Roll Control of a Small Aircraft

Performance Characteristics of Systems with Feedback

  1. First Order System Step Response
  2. Using Feedback to Modify Transient Response of a First Order System
  3. Characteristics of Under-damped, Second Order System Step Response
  4. Second Order System Step Response – Theoretical Analysis
  5. Second Order System Step Response – Summary
  6. Dominant (or Insignificant) Poles
  7. System Type and Steady-State Error
  8. Performance Indices
  9. Performance Examples
  10. Performance Design Problem – Aircraft Roll Control

Stability Analysis using the Routh-Hurwitz Method

  1. Routh-Hurwitz Criterion
  2. Design Problem – Rocket Attitude Angle Control (Single Degree of Freedom)

Closed-loop System Analysis using Root Locus Diagrams

  1. What is a Root Locus Diagram?
  2. Introduction to Root Locus Diagrams
  3. Root Locus Diagrams
  4. Summary Procedure for Root Locus Diagrams
  5. Root Locus Diagram Examples
  6. Using MATLAB for a Root Locus Analysis

Closed-loop System Analysis using Bode Diagrams

  1. Introduction to Bode Diagrams
  2. Summary of Bode Diagram Terms
  3. Bode Diagram of a 2nd Order System
  4. Bode Magnitude Diagram Examples
  5. Blank Bode Magnitude Diagram (for practice sketches)
  6. Frequency Domain Analysis

Control of Spring-Mass-Damper Position using PID Control

PID Control of a Spring-Mass-Damper Position

Implementation of PID Controllers using Operational Amplifiers

Implementation of PID Controllers in Electronic Circuits

Introduction to State Space Methods

  1. Introduction to State-Space Models
  2. Introduction to State-Space Models and Transfer Functions

Exercises (all answers provided)

  1. Exercises #1 – Conceptual Block Diagrams
    Exercises #1 – Conceptual Block Diagrams – Answers
  2. Exercises #2 – Mathematical Models and Linearization
  3. Exercises #3 – Laplace Transforms and Ordinary Differential Equations
  4. Exercises #4 – Mathematical Models and Transfer Functions
  5. Exercises #5 – Block Diagram Reduction
  6. Exercises #6 – Transfer Functions from Closed-Loop Equations
    Exercises #6 – Transfer Functions from Closed-Loop Equations Answers
  7. Exercises #7 – Parametric Sensitivity
  8. Exercises #8 – Closed-Loop, Second-Order System Step Response
  9. Exercises #9 – Dominant and Insignificant Poles
  10. Exercises #10 – Steady-State Error
  11. Exercises #11 – Performance Indices
  12. Exercises #12 – Disturbance Response
  13. Exercises #13 – Routh-Hurwitz Stability Analysis
  14. Exercises #14 – Root Locus Diagrams
    Exercises #14 – Root Locus Diagrams Answers
  15. Exercises #15 – Bode Magnitude Diagrams
    Exercises #15 – Bode Magnitude Diagrams Answers
  16. Exercises #16 – State-Space Models and Transfer Functions

Equation Sheets

  1. Basic Laplace Transform Table
  2. Equation Sheet #1
  3. Equation Sheet #2
  4. Equation Sheet #3 – Step Response Plots