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Q1:16.10 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?See Answer
Q2: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) = 8See Answer
Q3:Problem 2 (15 points) Solve the differential equation below for y(t) using partial fraction expansion and inverse Laplace transforms. Show all steps for your work. d'y d³y d'y dy +11. +42 +64 +32y = S(t) dt4 dt ³ dt² dt d³y d'y dy -(0) dt dt dt = (0) = -(0)=y(0)=0See Answer
Q4: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)See Answer
Q5:Problem 4 (15 points) For the time function x(t) below write the expression for x(t) and give the Laplace transform X(s) 10 x(t) 50 5 L 10 15 Time (s) 20 25 30See Answer
Q6:Simulink Lab #1 Solve the equations below using Simulink over the interval 0 ≤t ≤50 and plot the values of x & y versus time on the same plot. Save your plot in a word or pdf file and upload it along with the Simulink file to the drop box. d² y dt² 2 d²x 2 dt² y(0) = 2 dy dt + 3y dy dt +2 dx dt -(0) = −1 +6y-2x = 0 + 2xy = 14 x(0) = 3 dx dt (0)=1See Answer
Q7: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.See Answer
Q8: MECH 350 Introduction to Mechatronics Lab 1: Step response of RC circuit system: Experimental and Simulation Analysis Objectives The objectives of this lab are to 1) Introduce students to experimental analysis and study of first-order RC circuit dynamic system. 2) Perform step response on the first-order RC circuit system through an experiment in the lab. 3) Perform step response on the first-order RC circuit system through a computer simulation using MATLAB and SIMULINK. Introduction: Computer simulation and experimental study are the main methods used by scientists and engineers to verify, study and analyze dynamic systems, in addition to the analytical solution. In this lab, an RC circuit model will be used to study step response of such representative first order dynamic system, which was analytically introduced in the lecture. The study will be performed both experimentally and via simulation using MATLAB and SIMULINK. Consider the RC circuit shown in Figure 1. In the lecture the following first order differential equation was derived for this circuit. Vin = vo + RC Vin R ww dvo dt Vout Figure 1: An RC Circuit (1) Task 1: Experimental step response of RC circuit Instruments Used in this task: ● Function Generator ● Oscilloscope Measure and record the value of the components given (including the resistor and capacitor). Construct the RC circuit as shown in Figure 1 using a breadboard. The input signal Vin which comes from the function generator that will be set to output a square wave with a frequency of 50.0 Hz, an amplitude of 1.0 V and a DC offset of 0.5 V. This will result in a square signal with a min value of 0 and a max value of 1 V. This signal represents a repeating unit step function every 20 milliseconds. Display both the input and output signals on the oscilloscope Channel 1 & 2. You need to use a splitter on the function generator output. Collect and store the data from the oscilloscope on the desktop. Use the collected data to plot the response using MATLAB and to determine the steady state amplitude of the output signal, the gain and the time constant. Task 2: MATLAB Simulation of step response of RC circuit Use the MATLAB simulation function 'ode45' to simulate the system. You will need first to write a MATLAB function that compute the derivative term from Equation (1) assuming that the input vin and the output vo are known. Let us call the derivative term dvodt and use it as the output of the fucntion. In the function you need to define vin and give it a value and to use it with the value of vo that will be supplied as input to the function with the time t. function dvodt = 10e3; 0.1e-6; 1; с = vin dvodt Components Used in this task: Capacitor C=0.1 µF (Qty:1) Resistor R=10 KN (Qty:1) Breadboard = = = RC circuit (t, vo) 1/ (R*C) * (vin >> [t, y]=ode45('RC_circuit',[0 .01], 0); Now you are ready to plot the result. >> plot(t,y) - vo); Assuming that we saved the function using the name "RC_circuit.m”, we can use the MATLAB built-in function "ode45", as indicated, below to generate the simulation data y and t. Use the plotted response to determine the steady state amplitude of the output signal, the gain and the time constant. Task 3: SIMULINK Simulation of step response of RC circuit Here we will use the same function 'ode45' but through the graphical simulation interface SIMULINK. There are a number of methods to do the simulation in SIMULINK. However, we will use the method that utilize the transfer function block. To do that convert the differential equation given by Equation (1) into a transfer function that relate Vo(s) to Vin(s). RC circuit_simulation - Simulink classroom use Model Browser SIMULATION FILE LIBRARY PREPARE a w↑ □ Ready DEBUG Stop Time 0.01 Normal Fast Restart RC circuit_simulation MODELING RC_circuit_simulation Step Back ▾ SIMULATE R*C₁s+1 out.time FORMAT 100% Run vo(s) Vin (s) APPS out.y = 1 RC s +1 X Step Stop REVIEW Forward auto(ode15s) Property Inspector (1) Figure 2: SIMULINK simulation model of RC circuit with a step response Build the SIMULINK model shown in Figure 2 and use it to simulate the response for 0.01 seconds. Task 4 Analysis Plot the response as given by the analytical solution for t = 0 to t = 0.01 seconds. Comment on the results obtained from the four tasks. To what extent the results match the analytical predictions, in terms of time constant and steady state amplitude of the output signal from the RC circuit.See Answer
Q9:/ Consider a homogeneous, rigid cube of side = 30 cm and density p t = 0 its attitude and angular velocity are: (0) = [50, 30, 60] deg, = 1000 kg/m³. Knowing that at 3/I (0) = [200, 30, 10] deg/s, 3/I create a SIMULINK model to find ⑱(t), (t) and B/T (t), for t = (0,60) s, knowing that the external toque acting on the cube is equal to: a) MB(t) = 102.[-6p(t), 0, -5r(t)] N. m, b) MB(t) = 102. [-5,0,0] N · m ₁₁ B/I 63 Б2 61 where p(t), q(t), and r(t) are the components of B/I (t). Fig.1 Cube's initial attitude and angular velocity. 12 • Comment on your results for both case a) and b) The kinematics must be written using the quaternions parametrization. Please include all plots in one figure (4-by-3 subplots). Submit your code and a screenshot of the plots.See Answer
Q10:7.3 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.See Answer
Q11: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.See Answer
Q12: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).See Answer
Q13: 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).See Answer
Q14:12.6 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.See Answer
Q15:5- (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. See Answer
Q16:16.3 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.See Answer

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