Assignment 1: System Modeling

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1 Assignment 1: System Modeling Problem 1. (10 pts.) Consider a biological control system consisting of a human reaching for an object. Below is a list of general block diagram elements (on the left, labeled a-g) and elements for this specific example (on the right). General block diagram elements Elements for the human-reaching example (in order) a. process arm dynamics b. disturbance gravity c. actuator muscles d. sensor eyes and proprioceptors* e. controller brain f. reference signal object position g. output signal hand position *Definition of proprioceptors: sensors that provide information about joint angle, muscle length, and muscle tension, which are integrated to give information about the position of the limb in space. First, match each general block diagram element to the correct element for the human-reaching example. Second, draw the feedback control block diagram (similar to the examples from Lecture 1) for human reaching. It must contain all of the elements listed on the right above, each labeled with the letter corresponding to its general function in the block diagram (i.e., the matches that you made for the first part of the problem). Also, label the (h) error signal, (i) control signal, and (j) plant. Note: 0.5 points for each of 10 lettered items identified (a through j) for a total of 5 points, and 5 points for correct block diagram.

2 Problem 2. (10 pts.) Consider a system of learning that involves three blocks: a teacher, students, and examinations (which are used to measure the learning of the students). Draw a feedback control block diagram for this system. Identify and label on the block diagram possible elements (specific to this system) that could represent each of the following general block diagram elements: a. process students b. disturbance videos games (many examples here J ) c. sensor examinations d. controller teacher (in an ideal world!) e. reference signal desired knowledge (described in terms of an ideal exam score) f. control signal lectures or other means of communicating information to students g. output signal knowledge Note: 1 point for a reasonable choice for each element (total of 7), and 3 points for correct block diagram structure.

3 Problem 3. (10 pts.) The mass m in the figure below is attached to a rigid lever having negligible mass and negligible friction in the pivot at point O. The system s input is linear displacement x. When x and θ are zero, the springs are at their free length (i.e., they exert no force). Assume that both springs stay horizontal at all times. Neglect gravity. a. Derive the equation of motion for this system, with θ as the dependent variable and x as the input. (7 pts.)

4 b. Find the linearized equation of motion (i.e., assuming that θ is small). (1 pt.) c. What is the order of this differential equation? (1 pt.) Second order, since the dependent variable has a highest derivative of 2. d. The natural frequency of a system of this type is the square root of the equivalent stiffness divided by the equivalent mass. What is the natural frequency of the linearized system? (1 pt.)

5 Problem 4. (10 pts.) a. Assume R, C, L and v!" are all known. Find a single differential equation in terms of v!, with no other unknowns. (The labeled loops and node are hints!) (8 pts.) L + _ v L i L a v o v in _ + v R_ + R C 1 2 i R i C b. Find the Laplace transform of the differential equation (assume zero initial conditions) and compute the transfer function V! (s)/v!" (s). (2 pts.) Problem 5. (10 pts.) Consider the one-degree-of-freedom robot arm shown below. It has inertia J and

6 torsional damping b. A motor (whose dynamics you can ignore) applies a torque τ at the center of rotation. The angle of the arm is given by θ. Neglect gravity. a. Develop the equation of motion for this plant. (3 pts.) b. Is the system described by a first-order or second-order differential equation? (1 pts.) Second order if you consider the dependent variable θ, first order if you considered ω = θ. c. Take the Laplace Transform of the equation of motion. Call the transformed angle Θ(s) and the transformed torque Τ(s). Include initial conditions. (4 pts.) d. Assume zero initial conditions, then obtain the transfer function from torque input to angle output, which is Θ(s)/Τ(s). (2 pts.)

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