ME 475: Introduction of Mechatronics Assignment 6 Question 1 [Marks: 15] Problems 11 in Page 145 of the book of Control System Engineering (Fifth Edition). For each system shown in following Figure, write the state equations and the output equation for the phase-variable representation. (a)

𝐺(𝑠) =5𝑠 + 10

𝑠4 + 2𝑠3 + 𝑠2 + 5𝑠 + 10

(b)

𝐺(𝑠) =𝑠4 + 2𝑠3 + 12𝑠2 + 7𝑠 + 3𝑠5 + 9𝑠4 + 10𝑠3 + 8𝑠2

Question 2 [Marks: 15] Problems 4 in Page 682 of the book of Control System Engineering (Fifth Edition). Given the following open-loop plant, [Section: 12.2]

Design a controller to yield a 15% overshoot and a settling time of 0.75 second. Place the third pole 10 times as far from the imaginary axis as the dominant pole pair. Use the phase variables for state-variable feedback. Question 3 [Marks: 25] The transfer function of a human leg relates the output angular rotation about the hip joint to the input torque supplied by the leg muscle. A simplified model for the leg is shown in the following figure. The model assumes an applied muscular torque, Tm(t), viscous damping, D, at the hip joint, and inertial, J, around the hip joint. Also, a component of the weight of the leg, Mg, where M is the mass of the leg and g is the acceleration due to gravity, creates a nonlinear torque. If we assume that the leg is of uniform density, the weight can be applied at L/2, where L is the length of the leg. Do the following: (a) Set up the differential equation for this nonlinear system. (b) Find the transfer function, δθ(s) /Tm (s) , where δθ is the small angular rotation of the leg about the hip joint. (c) Use SIMULINK to simulate the step responses based on the results obtained in (a) and (b), respectively, given that J = 0.26 kg⋅m2, D = 1.0 N⋅m⋅s/rad, M = 2.5 kg, and L = 0.4 m. (Please hand in your SIMULINK model and simulation results)

Question 4 [Marks: 15] Find the linearized transfer function, G(s) = V(s)/I(s), for the electrical network shown in the following figure. The network contains a nonlinear resistor whose voltage-current relationship is defined by ir =ev2. The current source, i(t), is a small signal generator.

Question 5 [Marks: 30] Use SIMULINK to do the simulations of the room or furnace temperature control system and the web tension control system. The description of both systems and their SIMULINK models are available in Pages 597- 600 of the book: S. Cetinkunt, Mechatronics. (Please hand in your SIMULINK model and simulation results)