EXERCISE 4.17.16: (Chapter 4, Problem 16 in the 8th Edition). Find the location of the poles of second-order systems with the following specifications: [Section: 4.61 (a) %OS = 15; T₂ = 0.5 second (b) %OS = 8; T₂ = 10 seconds (c) Ts = 1 second; Tp = 1.1 seconds

1. Convert the following transfer function to state-space form. Assume zero initial conditions. For full credit, write the differential equation, show the new variables, write the state-space form, and then separately write out each of the matrices and vectors separately from the state-space form.

2. For the second-order system that follows, find the damping ratio, natural frequency, 2% settling time, peak time, rise time, and percent overshoot.

For this discussion, you will work with your group to complete the following task: Give an example of a second-order system from daily life and discuss its important performance specifications.

Question (4) For the control system shown in Fig.4 (a) Discuss the stability of the closed-loop system as a function of K by applying the Routh's stability criterion. (b) Determine the value of K which will cause sustained oscillations in the closed-loop system. (c) What are the corresponding oscillation frequencies.

Computer Assignment 1 When submitting the solution, follow these instructions: o Include a printout of your code, any related figures, and any handwritten work. o Do NOT email your code to the instructor 1: Plot the step responses for the following Transfer Function (Hint: use the "step" function and "stepinfo" functions in Matlab). Label the rise time, 2% settling time, peak time and percent overshoot on the plots. Also confirm these values using the methods taught in lecture (show your work). For full credit, do the following: o Plot the response; label the axes. o Mark out the time points/segments on the x-axis and any required points on the y-axis that correspond to the performance measures (rise time, 2% settling time, peak time, and percent overshoot). o Write the values for each performance measures from the plot. o Show the computations using the formulae from the lecture. o Confirm the values match.

Computer Assignment 2 1. Given the system below: 2. Given the system below: Write a program in Matlab for the problem above that does the following (provide printouts of your code, root locus plot, damping ratio lines, and step response): a- Display the root locus and pause (use command “rlocus”) b- Display a close-up of the root locus where the axes go from -2 to 2 on the real axis and -2 to 2 on the imaginary axis (use the commands “subplot” and “axis”) c- Overlay the 0.707 damping ratio line on the close-up root locus. (use command “sgrid”) d- Allow yourself to select interactively the point where the root locus crosses the 0.707 damping ratio line and give yourself the gain at that point (use command “rlocfind”) e- Generate the step response at the gain for 0.707 damping ratio (use command “step”)

Take the Laplace transform of the following set of differential equations and find the transfer function, G(s), connecting u(s) and y(s), y = Gu

1. Tune a PID controller for the following first-order process: Gain = 1.8 Deadtime = 1.5 minutes First time constant = 4.0 minutes Find the PI and PID controller parameters using the four open-loop methods (Ziegler-Nichols, Cohen- Coon, Cohen-Coon with constraints, and Fertik). For the Fertik method, tune for a setpoint response. 2. Tune a PID controller for the following second-order process: Gain = 1.0 Deadtime = 3.0 minutes First time constant = 2.0 minutes Second time constant = 4.0 minutes On the next page is a plot from which to determine: T₁, T2, T3, ACV, APV, 8, T, and K. The method is similar to the example on the Chapter 10 PowerPoint slides pages 10-45 to 10-48, that is, find the approximate first-order model of the process. Using the constants obtained above, find the PI controller parameters using the four open loop methods (Ziegler-Nichols, Cohen-Coon, Cohen-Coon with constraints, and Fertik). For the Fertik method, tune for a disturbance response.

2 Find the following: a. The range of K that keeps the system stable. b. The value of K that makes the system oscillate. c. The frequency of oscillation when K is set the value that makes the system oscillate.