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

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

**Notes**

**Knowledge Map**

Map of Areas Related to Control Systems

**Introduction to Control Systems**

**Introduction to Mathematical Modeling of Dynamic Systems**

- Mathematical Models
- Linearity Principles
- Linearization of Algebraic Models
- Linearization of Equations of Motion of Pendulum and Accelerating Pendulum
- Model for a Hydraulic Positioning System

**Short Review of Laplace Transforms**

- Using Laplace Transforms to Solve Linear, Ordinary Differential Equations
- Laplace Transform Table
- Laplace Transform Examples
- Introduction to Partial Fraction Expansions
- Examples using Partial Fraction Expansions

**Introduction to Transfer Functions and Block Diagrams**

- Transfer Functions
- Cramer’s Rule for Solving a System of Linear Equations
- Block Diagrams and Transfer Functions
- Block Diagrams & Transfer Functions Example
- Block Diagram Reduction Transformations
- Block Diagram Reduction Examples
- Armature-Controlled DC Motor Transfer Function

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

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

**General Characteristics of Systems with Feedback**

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

**Performance Characteristics of Systems with Feedback**

- First Order System Step Response
- Using Feedback to Modify Transient Response of a First Order System
- Characteristics of Under-damped, Second Order System Step Response
- Second Order System Step Response – Theoretical Analysis
- Second Order System Step Response – Summary
- Dominant (or Insignificant) Poles
- System Type and Steady-State Error
- Performance Indices
- Performance Examples
- Performance Design Problem – Aircraft Roll Control

**Stability Analysis using the Routh-Hurwitz Method**

**Closed-loop System Analysis using Root Locus Diagrams**

- What is a Root Locus Diagram?
- Introduction to Root Locus Diagrams
- Root Locus Diagrams
- Summary Procedure for Root Locus Diagrams
- Root Locus Diagram Examples
- Using MATLAB for a Root Locus Analysis

**Closed-loop System Analysis using Bode Diagrams**

- Introduction to Bode Diagrams
- Summary of Bode Diagram Terms
- Bode Diagram of a 2nd Order System
- Bode Magnitude Diagram Examples
- Blank Bode Magnitude Diagram (for practice sketches)
- 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**

**Exercises **(all answers provided)

- Exercises #1 – Conceptual Block Diagrams

Exercises #1 – Conceptual Block Diagrams – Answers - Exercises #2 – Mathematical Models and Linearization
- Exercises #3 – Laplace Transforms and Ordinary Differential Equations
- Exercises #4 – Mathematical Models and Transfer Functions
- Exercises #5 – Block Diagram Reduction
- Exercises #6 – Transfer Functions from Closed-Loop Equations

Exercises #6 – Transfer Functions from Closed-Loop Equations Answers - Exercises #7 – Parametric Sensitivity
- Exercises #8 – Closed-Loop, Second-Order System Step Response
- Exercises #9 – Dominant and Insignificant Poles
- Exercises #10 – Steady-State Error
- Exercises #11 – Performance Indices
- Exercises #12 – Disturbance Response
- Exercises #13 – Routh-Hurwitz Stability Analysis
- Exercises #14 – Root Locus Diagrams

Exercises #14 – Root Locus Diagrams Answers - Exercises #15 – Bode Magnitude Diagrams

Exercises #15 – Bode Magnitude Diagrams Answers - Exercises #16 – State-Space Models and Transfer Functions