1. A very strange input is described by the time-domain expression u(t) = -4sin(t)H(t)+5(t5)H(t5)-5(t-10)H(t-10)+4sin [(t 10)] H(t-10) (a) Sketch this input (b) What is the time-domain piece-wise function that describes this input? (c) What is the Laplace transform of this input? 2. A first order system with a time delay has the transfer function G(s) = Ke-as TS+1 This process is subjected to an input shown below. Obtain the response, y(t) of this process in response to the input below u(t) A a b t 3. The deviation temperature, y(t), on a critical tray in a pyrolysis fractionator is to be controlled (as indicated in the figure below) by employing a feedforward controller the manipulates the underflow reflux flow rate, u(t), on the basis of a measure of any devia- tions in the feed flow rate d(t). By experimental testing, the following model describes the system Y(s) G(s)U(s) + GD(s)D(s) = -5 G(s) = 6s+1 20 Gp(s) = 30s+1 (a) Obtain an expression for the dynamic response of the tray temperature, y(t), to a unit step in the reflux flow rate given the initial temperature of the tray is 250F and that the feed flow rate remained constant. To what ultimate value will the temperature settle after this unit change? (b) Obtain an expression for the dynamic response of the tray temperature, y(t), to a unit step in the feed flow rate given the initial temperature of the tray is 250F and that the reflux flow rate remained that same. To what ultimate value will the temperature settle after this unit change? (c) Using Simulink, plot the dynamic responses described in parts A and B. Provide printouts of the system response and of the Simulink model (please annotate the model to show values of inputs, etc.) Do these figures agree with your expressions from parts A and B? Which variable affects the tray temperature the most (in terms of the absolute change in the tray temperature) and which affects it the quickest? (d) Using your solutions from parts A and B, what would be the ultimate value of the temperature when a unit step is implemented in both the feed flow rate and reflux flow rate? Feed D PYROLYSIS FRACTIONATOR Y Underflow Reflux U Feedforward Controller 4. CHALLENGE PROBLEM: The transfer function from question 2 is subjected to the input from question 1. K = 5, 7 = 10, and a = 2. (a) What is the final value of the process in response to this input? (b) Approximately, how long after the input starts does the response does the system reach 99% of the final value of process? Note: Very little calculation should required. (c) What is the time domain function that describes this response? (d) Plot this response. (Note: A plot is a detailed figure that should be created on a computer) 5. EXTRA CREDIT: You are preparing to model how a process unit you are in charge of responds to changes in ambient temperature. Determine the Laplace transform of an input the replicates how the temperature in Pullman, WA varies on an hourly basis throughout the year. State all assumptions including the date and time that your input begins. Cite relevant data if necessary. A solution will earn no credit if it is not justified, even if correct.