Homework 3

Learning Objectives

• Introduce basic controls concepts.
• Connect the ideas of control systems directly to mechanical systems.
• Deepen the understanding of the Mass-Spring damper control system from lecture.

P1. Creating Equations from Mechanical Systems(10 points) - Individual

In the design and analysis of control systems, we model the control process through various mathematical representations. One of such representations is the linear time-invariant systems (LTI) - a dynamic system description by linear, constant-coefficient, differential equations.

As we've seen in lecture, we will be converting a mass-spring damper system into a LTI differential equation. For supplemental instruction, see the pdf at this link.

In the mechanical system diagram below, m is the mass, k is the spring constant, b is the friction constant, u(t) is the external applied force, and y(t) is the resulting displacement.

Similar to the derivation in lecture, we can translate this system into a system of equations with respect to time. This is called the "time-variable differential equation" used to describe the mechanical system. We've derived the equation of the system (from Newton's second law as F=ma) as follows:

1. Using the "Laplace transform" as shown in the supplemental instruction.
2. Then, using your equation, create the transfer function for the Mechanical System. Please show your work.

P2. Creating Control System Block Diagrams from Equations(5 points) - Individual

1. Using your equation describing the mechanical system in P1, create a block diagram as seen in the lecture notes. You can find supplemental instruction at this link
2. What kind of controller does this mechanical system depict? (Hint: Check the titles of the lecture notes)

P3. Closed Loop Control Systems(10 points) - Individual

Sketch and briefly describe some graphs like the one about showing the response for a PID controller with:

• Too large kP
• Too small kP
• Too large kI
• Too small kI
• Too large kD
• Too small kD

L1. Lab 3: Dead Reckoning (75 points) - Individual

To receive any credit on the lab, you must complete the feedback form here: Form.

Upon completing the form, you will receive a passphrase. Please submit in your PDF.

Submission:

Collect your answers in a PDF named [AndrewID]_hw3.pdf, where [AndrewID] is replaced by your Andrew ID, minus the brackets. For example, user with AndrewID odb would submit odb_hw3.pdf. Instructions to submit the pdf will be on Piazza. Failure to follow instructions on this homework will result in an automatic zero.