 # ME401: Dynamic Systems & Controls

Unit 1: Dynamic Systems   Unit One will introduce dynamic systems.  Dynamic systems are mathematically modeled as differential equations representing various components and the interactions between them.  As all dynamic systems are represented mathematically, this unit will first review differential equations and Laplace transforms matrices in order to help you better understand controller design, which we will cover in subsequent chapters.  After we have completed our math review, we will learn how to represent dynamic systems in various forms.  You will discover that the analysis of dynamic systems is usually performed in either time-domain or frequency-domain.  You will also learn how to represent dynamic systems in transfer function using Laplace transforms.

Unit 1 Time Advisory
This unit should take you 32 hours to complete.

☐    Subunit 1.1: 12 hours

☐    Subunit 1.1.1: 1 hour

☐    Subunit 1.1.2: 1 hour

☐    Subunit 1.1.3: 2.5 hours

☐    Subunit 1.1.4: 2 hours

☐    Subunit 1.1.5: 2 hours

☐    Subunit 1.1.6: 2 hour

☐    Subunit 1.1.7: 1.5 hours

☐    Subunit 1.2: 11 hours

☐    Subunit 1.2.1: 3 hours

☐    Subunit 1.2.2: 3 hours

☐    Subunit 1.2.3: 2.5 hours

☐    Subunit 1.2.4: 2.5 hours

☐    Subunit 1.3: 4 hours

☐    Subunit 1.3.1: 2 hours

☐    Subunit 1.3.2: 2 hours

☐    Subunit 1.4: 5 hours

☐    Subunit 1.4.1: 1 hour

☐    Subunit 1.4.2: 1.5 hours

☐    Subunit 1.4.3: 1.5 hours

☐    Subunit 1.4.4: 1 hour

Unit1 Learning Outcomes
Upon successful completion of this unit, students will be able to:

• Define dynamic systems and types.
• Identify how mechanical, thermal, fluid, and electrical systems are modeled.
• Develop and review the required mathematical background for dynamic systems and control.
• Identify the characteristics of first- and second-order dynamic systems.

1.1 Dynamic Systems: Introduction and Modeling   1.1.1 Introduction to Dynamic System and Controls.   - Reading: Georgia Tech University: Dr. Sadegh’s “Introduction to System Dynamics and Control” Link: Georgia Tech University: Dr. Sadegh’s “Introduction to System Dynamics and Control” (PDF)

Instructions: Please click and read the PDF titled “Introduction to System Dynamics and Control” under “Selected Lecture Notes.”

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displayed on the webpage above.
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1.1.2 Examples of Dynamic Systems   - Reading: Bucknell University: Dr. Mastascusa’s “An Introduction to System Dynamics—First Order Systems: The System and Some Examples” Link Bucknell University: Dr. Mastascusa’s “An Introduction to System Dynamics—First Order Systems: The System and Some Examples” (HTML)

Instructions: Please click the link, open the webpage, and read the section titled “The System—And Some Examples.”

1.1.3 Mechanical Systems   - Reading: Georgia Tech University: Dr. Sadegh’s “Mathematical Modeling of Mechanical Systems” Link: Georgia Tech University: Dr. Sadegh’s “Mathematical Modeling of Mechanical Systems” (PDF)

Instructions: Please click and read the PDF titled “Mathematical Modeling of Mechanical Systems” under “Selected Lecture Notes.”

1.1.4 Fluid Systems   - Reading: Georgia Tech University: Dr. Sadegh’s “Mathematical Modeling of Fluid Systems” Link: Georgia Tech University: Dr. Sadegh’s “Mathematical Modeling of Fluid Systems” (PDF)

Instructions: Please click and read the PDF titled “Mathematical Modeling of Fluid Systems” under “Selected Lecture Notes.”

1.1.5 Electrical Systems   - Reading: Georgia Tech University: Dr. Sadegh’s “Mathematical Modeling of Electrical Systems” Link: Georgia Tech University: Dr. Sadegh’s “Mathematical Modeling of Electrical Systems” (PDF)

`````` Instructions: Please click and read the PDF titled “Mathematical
Modeling of Electrical Systems” under “Selected Lecture Notes.”

displayed on the webpage above.
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1.1.6 Linear Dynamic Systems   - Reading: University of California at Berkeley: Dr. Hedrick’s “General Properties: Properties of Linear Time Invariant Systems” Link: University of California at Berkeley: Dr. Hedrick’s “General Properties: Properties of Linear Time Invariant Systems” (PDF)

Instructions: Please click and open the PDF titled “General Properties” and then scroll down and read the section “Properties of Linear Time Invariant Systems.”

1.1.7 Non-Linear Dynamic Systems   - Reading: University of California at Berkeley: Dr. Hedrick “Nonlinear System Properties” Link: University of California at Berkeley: Dr. Hedrick “Nonlinear System Properties” (PDF)

Instructions: Please click and open the PDF titled “General Properties” and then scroll down and read the section “Nonlinear System Properties.”

1.2 Mathematical Background   Note: Once you understand different types of dynamic systems, we will review the mathematics that we will use throughout this course.  We will take a look at differential equations, Laplace transforms, matrices, and linearization of non-linear dynamic systems.  This review will help us better understand how to convert dynamic systems from one form of mathematical representation to another.  This section is optional.  You can go to section 1.3 if you are already familiar with the math.

1.2.1 Differential Equations   - Lecture: Khan Academy: Differential Equations: “Introduction to Differential Equations,” “Separable Differential Equations,” “First Order Homogenous Equations,” and “Second Order Homogenous Equations” Link: Khan Academy: Differential Equations: “Introduction to Differential Equations,” (YouTube) “Separable Differential Equations,” (YouTube) “First Order Homogenous Equations,” (YouTube) and “Second Order Homogenous Equations” (YouTube)

Also available in:
iTunes U (Introduction to Differential Equations)
iTunes U (Separable Differential Equations)
iTunes U (First Order Homogenous Equations)
iTunes U (Second Order Homogenous Equations)

Instructions: Please watch these videos to familiarize yourself with differential equations and relevant examples.

`````` Watching these videos should take approximately 45 minutes.

Attribution-NonCommercial-ShareAlike 3.0 United States
are attributed to the Khan Acadamey.
``````

1.2.2 Laplace Transforms   - Lecture: Khan Academy: Differential Equations: “Laplace Transform,” Laplace Transform to Solve an Equation,” and “Inverse Laplace Examples” Link: Khan Academy: Differential Equations: “Laplace Transforms” (YouTube), “Laplace Transform to Solve an Equation” (YouTube), and “Inverse Laplace Examples” (YouTube)

Also available in:
iTunes U (Laplace Transforms)
iTunes U (Laplace Transforms to Solve an Equation)
iTunes U (Inverse Laplace Examples)

Instructions: Please watch these videos to familiarize yourself with Laplace transforms and take a look at some relevant examples.

`````` Watching these videos should take approximately 45 minutes.

Attribution-NonCommercial-ShareAlike 3.0 United States
are attributed to the Khan Academy.
``````

1.2.3 Matrices   - Lecture: Khan Academy: Linear Algebra: “Introduction to Matrices” and “Matrices to Solve Systems of Equations” Link: Khan Academy: Linear Algebra: “Introduction to Matrices” (YouTube) and “Matrices to Solve Systems of Equations” (YouTube)

Also available in:
iTunes U (Introduction to Matrices)
iTunes U (Matrices to Solve Systems of Equations)

Instructions: Please watch these videos to familiarize yourself with matrices and to take a look at some examples.

`````` Watching these videos should take approximately 30 minutes.

Attribution-NonCommercial-ShareAlike 3.0 United States
are attributed to the Khan Academy.
``````

1.2.4 Transfer Functions   - Reading: University of Wisconsin-Madison: Dr. Duffie’s “Lecture 06: Transfer Functions” The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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1.3 Dynamic System Order and Types   Note: The order of a dynamic system is governed by the order of the differential equation representing the dynamic system.  Higher-order differential equations are more complex to solve; hence, one should use computational tools like SCILAB to solve them.

1.3.1 First-Order Dynamic Systems   - Reading: Bucknell University: Dr. Mastascusa’s “First Order System Descriptions” Link: Bucknell University: Dr. Mastascusa’s “First Order System Descriptions” (HTML)

Instructions: Please click the link and read the section titled “First Order System Descriptions.”

1.3.2 Second-Order Dynamic Systems   - Reading: Bucknell University: Dr. Mastascusa’s “An Introduction to Dynamics—Second Order System” Link Bucknell University: Dr. Mastascusa’s “An Introduction to Dynamics—Second Order System” (HTML)

Instructions:  Please click and read the entire webpage.

1.4 Characteristics of Second-Order Dynamic Systems   1.4.1 Impulse Response   - Reading: Bucknell University: Dr. Mastascusa’s “An Introduction to System Dynamics—Second Order Systems: Impulse Response” Link: Bucknell University: Dr. Mastascusa’s “An Introduction to System Dynamics—Second Order Systems: Impulse Response” (HTML)

Instructions: Please click the link and read the section titled “Impulse Response.”

1.4.2 Overshoot vs. Damping Ratio   - Reading: Bucknell University: Dr. Mastascusa’s “An Introduction to System Dynamics—Second Order Systems: Overshoot in Second Order Systems” Link: Bucknell University: Dr. Mastascusa’s “An Introduction to System Dynamics—Second Order Systems: Overshoot in Second Order Systems” (HTML)

Instructions: Please click the link and read the section titled “Overshoot in Second Order Systems.”