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.

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.

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).

12.3 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).

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.

5- (USE SIMULINK) The forcing function shown acts on a system whose model is: 3x + 6x + 1200x = f(t) Plot the forced response of the system.

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.