7.3 A process consists of two stirred tanks with input q and outputs T₁ and T₂ (see Fig. E7.3). To test the hypothesis that the dynamics in each tank are basically first-order, a step change in q is made from 82 to 85 L/min, with output responses given in Table E7.3. (a) Find the transfer functions T(s)/Q'(s) and T'(s)/T(s) Assume that they are of the form K₁/(t;s + 1). (b) Calculate the model responses to the same step change in q and plot with the experimental data.

16.3 Consider the cascade control system in Fig. E16.3. Use IMC tuning rules for both the master and slave controllers. Design K2 first, and then use that value to design G₁₁ (PI controller). The higher-order transfer function can be approximated first by a FOPTD model using a step test. Plot closed-loop responses for different values of the IMC closed-loop time constant for both outer loop and inner loop for a set point change.

Assignment The combination of all these elements produces a mathematical model for the Elbow Control System for the Robot Arm. Using this description as a basis, follow the steps outlined below to complete the first part of your assignment for this course: Mathematical Modelling & Continuous Time Simulation 1. Use the description given above to derive the state space model for the Robot Arm System. 2. Use this model and the parameter values given in the Appendix A to produce an equation or script based simulation of the Robot Arm System in Matlab. 3. Employ a suitable initial conditions and numerical integration solver with a suitable step-size in the simulation of your system. Justify your choice of the initial conditions, solver and step- size. Do not use the in-built Matlab integration functions. 4. Analyse the dynamic response of the system. Do you think this a good design for the Elbow Control System? Explain your answer. Block Diagram & Validation 5. Using basic, commonly used blocks in Simulink, construct a block diagram simulation of the Robot Arm System. 6. Use the responses from this block diagram simulation to validate your Matlab model from steps (1) & (2) and simulation responses from step (3).

12.6 Consider the FOPTD model in Eq. 12-10 with X= 5, T4, and 9 = 3. Design PI and PID controllers using the IMC tuning method with T, 3. Simulate the closed-loop systems for a unit step change in set point. Does the addition of derivative action result in significant improve-ment? Justify your answer.

PROJECT 2 For a two degree of freedom robot below that operates in a XY plane and in a vertical gravity field following is given: a) L1=L2=1m, b) links are massless, c) Ma = 1 kg is a point mass. Find the following. • Find the differential equations of motion of this robot. With this, you will obtain a system of two second order nonlinear equations. Inputs to your system are two torques that are acting on each joint. You may also assume that both joints have viscous damping. • Using a PID controller, form a closed loop configuration for each of the joints. • Implement such a controlled robot in Simulink. • Tune the PID controller so that each robot's joint has an aperiodic response for a step input. • Plot all of your results.

12.3 A process has the transfer function, G(s) = 2e-²/ (s + 1). Compare the PI controller settings for the fol- lowing design approaches: T2 (a) IMC method (t = 0.2) (b) IMC method (t = 1.0) (c) ITAE performance index (disturbance) (d) ITAE performance index (set point) (e) Which controller has the most conservative settings? Which has the least conservative? (f) For the two controllers of part (e), simulate the closed- loop responses to a unit step disturbance, assuming that G(s) = G(s).

In the case of the RLC circuit, the inductance L is 4 H and the capacitance C' is 1 F. In the case of the the mass is m is 4 kg and the spring constant k is 1 N/m. mass-spring-damper, 1. Choose one model. Using Simulink, set-up the appropriate block diagram correspond- ing to the governing differential equation (Equation 1 or 2). You'll also have to use a scope to monitor both the forcing function (either Vert or Fert and the output (either V or y. In the simulation, use a maximum step size of 0.01 (This is found in the "Model Configuration Parameters" under the simulation tab. If the maximum step size is left on auto, Simulink sometimes uses too large of a step and the outputs displayed in the Scope box will be jagged rather than smooth). For the integrator blocks, the initial condition is 0 at time 0. Run all simulations for 100 seconds. The appropriate boundary conditions are either y(0) = 0 and y'(0) = 0 or V(0) = 0 and V'(0) = 0. Finally, to introduce the forcing function into your block diagram use the "Step block" found under sources. Choose a step time of 10 s, an initial value of 0, and a final value of 10. If done correctly the forcing function will change from a value of 0 to 10 at 10 seconds./n2. Case 1: Run the simulation using either R = 0 or pf = 0N s/m. 3. Case 2: Run the simulation using either R = 0.62 or p = 0.6 N s/m. 4. Case 3: Run the simulation using either R= 1.52 or pf = 1.5N s/m. 5. Case 4: Run the simulation using either R = 32 or f = 3N s/m. 6. Case 5: Run the simulation using either R = 40 or y=4N s/m. 7. Case 6: Run the simulation using either R = 8N or pf = 8N-s/m. 8. Case 7: Run the simulation using either R = 16 or µ = 16 N. s/m. 9. Comment on your findings in Questions 2 through 8. In particular, you should find that Case 5 separates Cases 2-4 from Cases 6 and 7 based on the roots of the characteristic equation. Here, Cases 2-4 correspond to underdamped responses, Case 5 corresponds to a critically damped response, and Cases 6-7 correspond to overdamped responses. Case 1 corresponds to an undamped response. Here think about the influence of friction (or resistance) on the response. Do the trends seem appropriate in terms of what we understand about friction (or resistance)?/nDIRECTIONS: Number your Answers! Question 1: Show the block diagram that you used in Simulink with all blocks and signals clearly labeled. Questions 2-8: Show all 7 plots corresponding to each different value of either resistance or coefficient of friction. Question 9: Briefly discussion your findings in terms of the roots of the characteristic equation. In particular, you should find that Case 5 separates Cases 2-4 from Cases 6 and 7. Here, Cases 2-4 correspond to underdamped responses, Case 5 corresponds to a critically damped response, and Cases 6-7 correspond to overdamped responses. Case 1 corresponds to an undamped response. For each of these cases, calculate the damping ratio. Comment on the effect of the dammping ratio on the type of transient response. Save your assignment as a PDF and upload to the assignment submission window.

Problem 3 (5 points) A) Examine the feedback temperature control system in the Figure below and identify the following: Control Variable, Measured Variable, Manipulated Variable, Final Control Element, Sensor & Two Disturbance Variables B) Based on one of your disturbance variables explain how you could implement a feedforward control system Cooling medium 0000 (TC)

16.10 The closed-loop transfer function for the Smith predic- tor in Eq. 16-22 was derived assuming no model error. (a) Derive a formula for Y/Y, when G‡ G. What is the characteristic equation? (b) Let G=2e-2s/(5s +1). A proportional controller with Kc = 15 and a Smith predictor are used to control this process. Simulate set-point changes for ±20% errors in process gain (K), time constant (t), and time delay (six different cases). Discuss the relative importance of each type of error. (c) What controller gain would be satisfactory for ±50% changes in all three model parameters?

Problem 1 (15 points) Solve the nonlinear coupled differential equations using the default variable step solver in Simulink from 0 to 60 seconds and plot the response of the solutions x, y & z in a single labeled figure. Save the model file to the assignment drop box as pl.slx and print the labeled response plot including the legend along with the block diagram as part of your homework submission. Use dark colors for the plot line so they show up when printed. As part of your solution give the values of z, y and dy/dt and t=35 seconds. dz (10z - z²) d²y dt² + 25/y dx 2zx = 1 dt x (0) = 5 dy dt dt (0) = 3 = -0.5 ln(15z) dy dt + 4e¯y - 0.6y√√z = 0 y (0) = 1.5 z(0) = 8