# ME401: Dynamic Systems & Controls

Unit 2: Analysis of Dynamic Systems   Unit Two builds on the material covered in Unit One by asking you to analyze dynamic systems with respect to their responses to various input signals.  Systems can be subjected to constant or varying inputs.  Therefore, an understanding of how these systems react to various inputs is crucial to designing the appropriate controllers.  The characteristic equations of dynamic systems convey significant information regarding the system’s poles and zeros.  The response of a dynamic system in time and frequency domain is used to determine whether a system is stable, marginally stable, or unstable.  Once the system is analyzed for stability, appropriate controllers can be designed for the desired response.

This unit should take you 30 hours to complete.

☐    Subunit 2.1: 6 hours

☐    Subunit 2.1.1: 3 hours

☐    Subunit 2.1.2: 1 hour

☐    Subunit 2.1.3: 2 hours

☐    Subunit 2.2: 11 hours

☐    Subunit 2.2.1: 1 hour

☐    Subunit 2.2.2: 1.5 hours

☐    Subunit 2.2.3: 1.5 hours

☐    Subunit 2.2.4: 1.5 hours

☐    Subunit 2.2.5: 1 hour

☐    Subunit 2.2.6: 2.5 hours

☐    Subunit 2.2.7: 2 hours

☐    Subunit 2.3: 12 hours

☐    Subunit 2.3.1: 1.5 hours

☐    Subunit 2.3.2: 1.5 hours

☐    Subunit 2.3.3: 2 hours

☐    Subunit 2.3.4: 1.5 hours

☐    Subunit 2.3.5: 1.5 hours

☐    Subunit 2.3.6: 1.5 hours

☐    Subunit 2.3.7: 1.5 hours

☐    Subunit 2.3.8: 1 hour

Unit2 Learning Outcomes
Upon successful completion of this unit, students will be able to:

• Analyze dynamic systems in time-domain.
• Analyze dynamic systems in frequency-domain.
• Identify stability of dynamic systems for controller design.

2.1 Analysis and Response of Dynamic Systems in Time-Domain   2.1.1 Response of Systems   - Reading: Bucknell University: Dr. Mastascusa’s “An Introduction to System Dynamics—Second Order Systems: A Note on System Responses” Link: Bucknell University: Dr. Mastascusa’s “An Introduction to System Dynamics—Second Order Systems: A Note on System Responses” (HTML)

Instructions: Please click the link, scroll down, and read the section titled “A Note on System Responses.”

2.1.2 Steady State Error (SSE)   - Reading: Bucknell University: Dr. Mastascusa’s “Steady State Error in Control Systems: Control System Design: What is SSE?” Link: Bucknell University: Dr. Mastascusa’s “Steady State Error in Control Systems: Control System Design: What is SSE?” (HTML)

`````` Instructions: Please click the link, scroll down, and read the
section titled “What is SSE.”

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2.1.3 Root-Locus   - Reading: Bucknell University: Dr. Mastascusa’s “An Introduction to the Root Locus Techniques” Link: Bucknell University: Dr. Mastascusa’s “An Introduction to the Root Locus Techniques” (HTML)

2.2 Analysis and Response of Dynamic Systems in Frequency-Domain   2.2.1 Bode Plots   - Reading: Bucknell University: Dr. Mastascusa’s “Bode Plots: Why Bode Plots?” and “Bode Plots: What are Bode Plots?” Link: Bucknell University: Dr. Mastascusa’s “Bode Plots: Why Bode Plots?” (HTML) and “Bode Plots: What Are Bode Plots?” (HTML)

`````` Instructions: Please click the link and read the sections titled,
"Why Bode Plots?" and "What are Bode Plots?"

Please note that Bode's name, which is of Dutch origin, is
occasionally written with an accent at the end: Bode'.  The name is
pronounced *Boh-dee* in English.

displayed on the webpage above.
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2.2.2 Bode Plot of First-Order Systems   - Reading: Bucknell University: Dr. Mastascusa’s “Bode Plots: Bode Plots for First Order Systems” Link: Bucknell University: Dr. Mastascusa’s “Bode Plots: Bode Plots for First Order Systems” (HTML)

Instructions: Please click the link and read the section titled “Bode Plots for First Order Systems.” In this reading, Dr. Mastascusa introduces the concept of using Bode plots to analyze the frequency response of first-order dynamic systems.

2.2.3 Concept of Decibels   - Reading: Bucknell University: Dr. Mastascusa’s “Bode Plots: Decibels and More” Link: Bucknell University: Dr. Mastascusa’s “Bode Plots: Decibels and More” (HTML)

Instructions: Please click the link, scroll down, and read the section titled “Decibels and More.”  In this reading, Dr. Mastascusa introduces you to the concept of decibels and how decibels are used to analyze dynamic systems using frequency response.

2.2.4 Bode Plot of Second-Order Systems   - Reading: Bucknell University: Dr. Mastascusa’s “Bode Plots: Bode Plots for Second Order Systems” Link: Bucknell University: Dr. Mastascusa’s “Bode Plots: Bode Plots for Second Order Systems” (HTML)

Instructions: Please click the link, scroll down, and read the section titled “Bode Plots for Second Order Systems.”  In this reading, Dr. Mastascusa introduces you to Bode plots as used in analyzing the frequency response of second-order dynamic systems.

2.2.5 Bode Plot for Large Systems   - Reading: Bucknell University: Dr. Mastascusa’s “Bode Plots: Bode Plots for Larger Systems—Examples” Link: Bucknell University: Dr. Mastascusa’s “Bode Plots: Bode Plots for Larger Systems—Examples” (HTML)

Instructions: Please click the link and read the section titled “Bode Plots for Larger Systems—Examples.”  In this reading, Dr. Mastascusa introduces the use of Bode plots in analyzing the frequency response of large dynamic systems.

2.2.6 Nyquist Plots   - Reading: Bucknell University: Dr. Mastascusa’s “Nyquist Plots: What is a Nyquist Plot?” Link: Bucknell University: Dr. Mastascusa’s “Nyquist Plots: What is a Nyquist Plot?” (HTML)

Instructions: Please click the link and read the section titled “What is a Nyquist Plot?”  In this reading, Dr. Mastascusa introduces the use of Nyquist plots in analyzing the frequency response of dynamic systems.

2.2.7 Gain Margin   - Reading: Bucknell University: Dr. Mastascusa’s “Nyquist Stability: Gain Margin” Link: Bucknell University: Dr. Mastascusa’s “Nyquist Stability: Gain Margin” (HTML)

Instructions: Please click the link and read the section titled “Gain Margin.”  In this reading, Dr. Mastascusa introduces the concept of gain margin and how the concept can be used in analyzing dynamic systems using frequency response.

2.3 Stability of Dynamic Systems   Note: The location of the poles and zeros and the values of phase and gain margins have a bearing on a system’s stability.  Many researchers and scientists have discussed various stability criteria in both time and frequency domains.  SCILAB is a great tool for plotting and analyzing these systems.

2.3.1 Relative Stability   - Reading: Bucknell University: Dr. Mastascusa’s “Relative Stability” Link: Bucknell University: Dr. Mastascusa’s “Relative Stability” (HTML)

Instructions: Please click the link and read the entire webpage.  In this reading, Dr. Mastascusa introduces the concept of relative stability and explain how it can be used to analyze systems during controller design.

2.3.2 Bode Plots for Stability Determination   - Reading: Bucknell University: Dr. Mastascusa’s “Nyquist Stability: Using Bode Plots to Evaluate Stability” Link: Bucknell University: Dr. Mastascusa’s “Nyquist Stability: Using Bode Plots to Evaluate Stability” (HTML)

Instructions: Please click the link and read the entire webpage.  In this reading, Dr. Mastascusa introduces Bode plots and explains how they are used to evaluate relative stability in order to analyze systems during controller design.

2.3.3 Gain for Stability Using Bode Plot   - Reading: Bucknell University: Dr. Mastascusa’s “Determining Gain Value for Stability from a Bode Plot” Link: Bucknell University: Dr. Mastascusa’s “Determining Gain Value for Stability from a Bode Plot” (HTML)

`````` Instructions: Please click the link and read the section titled
“Determining Gain Value for Stability from a Bode Plot.”

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2.3.4 Phase for Stability Using Bode Plot   - Reading: Bucknell University: Dr. Mastascusa’s “Relative Stability: Interpreting Phase and Gain Margin on a Bode Plot” Link: Bucknell University: Dr. Mastascusa’s “Relative Stability: Interpreting Phase and Gain Margin on a Bode Plot” (HTML)

`````` Instructions: Please click the link and read the section titled
“Interpreting Phase and Gain Margin on a Bode Plot.”

displayed on the webpage above.
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2.3.5 Nyquist Stability Criterion   - Reading: Bucknell University: Dr. Mastascusa’s “The Nyquist Stability Criterion: Applying the Nyquist Stability Criterion” Link: Bucknell University: Dr. Mastascusa’s “The Nyquist Stability Criterion: Applying the Nyquist Stability Criterion” (HTML)

`````` Instructions: Please click the link and read the section titled
“Applying the Nyquist Stability Criterion.”

displayed on the webpage above.
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2.3.6 Gain for Stability Using Nyquist Plot   - Reading: Bucknell University: Dr. Mastascusa’s “Determining Gain Value for Stability from a Nyquist Plot” Link: Bucknell University: Dr. Mastascusa’s “Determining Gain Value for Stability from a Nyquist Plot” (HTML)

2.3.7 Phase Margin   - Reading: Bucknell University: Dr. Mastascusa’s “Relative Stability: Phase Margin” Link: Bucknell University: Dr. Mastascusa’s “Relative Stability: Phase Margin” (HTML)

`````` Instructions: Please click the link and open up the webpage read