Homework Answers
Therefore, the variation of E(y) with respect to x1= 10 and variation of E(y) with respect to x2= 9.
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(Problem 2: 10 points) Consider the following two-input second order model with interaction terms, E() 3...
Problem 5 Information Problem 6: 10 points A) Find moments of first and second order and...
Problem 5 Information
Problem 6: 10 points A) Find moments of first and second order and the variance for variable M in Problem 5 (B) Find expectation and variance for variable N in Problem 5. Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn} uniformly distributed over the unit interval, (0,1) Introduce two new random variables, M-max (Xi, X2,..., Xn) and N- min (X1, X2,... ,Xn)
Problem 1 Consider two first order low-pass systems connected in parallel: -2u The objective is to determine a second order ODE describing the variable y by manipulating the differential equations (n...
Problem 1 Consider two first order low-pass systems connected in parallel: -2u The objective is to determine a second order ODE describing the variable y by manipulating the differential equations (no transfer function techniques are allowed). Answer the following series of questions: 1. (2 points) Write the variable y in terms of i and 2 2. (6 points) Determine the relationship between y, j, and z1, z2 and u. Write your final expression in a matrix-vector format: ? 01 ??...
0 2 10 0 2 8 Consider the multiple regression model where є¡ ~ iid Ņ(0, σ*) for i = i, 2, 3, 4, 5...
0 2 10 0 2 8 Consider the multiple regression model where є¡ ~ iid Ņ(0, σ*) for i = i, 2, 3, 4, 5. (c) Fill in the values for the following ANOVA table: Source of Variation Sum of Squares df Mean Square F Regression on Xi, X2 Error Total (Corrected) (d) State the nul and alternative hypotheses associated with the F test from the ANOVA table in part (c) and do the F test (e) Compute R2 (f)...
Problem 2: Consider again the two RLC circuits from HW1 Problem 6 C L IKs IKs...
Problem 2: Consider again the two RLC circuits from HW1 Problem 6 C L IKs IKs L In HW1 Problem 6 you found the transfer function Vc(s)V(s) for each of the circuits, using Impedance Analysis. You essentially assumed zero initial conditions (for the capacitor's voltage S and for the inductor's current) 2.1. Develop a state-variable model for each of the circuits, where the state variables are (in both circuits) xi vc and x2 i That is, derive (for each circuit)...
3. Consider the following production problem Maximize 10r 12r2 20r, subject to the constraints xi...
3. Consider the following production problem Maximize 10r 12r2 20r, subject to the constraints xi +x2 +x3 10. ri + 2r2 +3rs 3 22, 2x1 2a2 +4x3 S 30 120, x2 20, 0 (a) (2 points) Solve the problem using the simplex method. Hint: Check your final tableau very carefully as the next parts will depend on its correct- ness. You will end up having 1, 2, r3 as basic variables. (b) (6 points) For1,2, and 3, determine the admissible...
• (Problem 2) Consider the second version of the Lotka-Volterra model: F(a – bF – cS)...
• (Problem 2) Consider the second version of the Lotka-Volterra model: F(a – bF – cS) dF dt ds dt S(-k + \F). (1) Explain the model; i.e. what are the terms in the equation signify? How is this model different from equations (1)? (2) Find the equilibrium point(s). (3) Linearize (2) about the equilibrium point(s). (4) Classify if the equilibrium points are stable or unstable. (5) Pick some values for a, b, c, k, 1. Plot the solutions of...
1. Consider the following model where y denotes the tool life and xi, x2, and tz denote the cutti...
1. Consider the following model where y denotes the tool life and xi, x2, and tz denote the cutting speed, tool type, and type of cutting oil, respectively. There are two different tool types, A and B, and there are two type of cutting oils, low-viscosity oil and medium-viscosity oil. The two categorical predictors are 1 if type A 1 f low-viscosity oil used defined as *20 if type B and c3 0 if medium-viscosity oil used (a) Interpret the...
Problem 4 (10 points) Astrom et a1 have presented a simplified second-order transfer function model for...
Problem 4 (10 points) Astrom et a1 have presented a simplified second-order transfer function model for bicycle dynamics as C1 Parameters defining the bicycle geometry. Schematic (a) top and (b) rear views of a bicycle The steer angle is δ and the roll angle is φ. The input is δ(t), the steering angle, and the output is Plt), the tilt angle (between the the bicycle longitudinal plane and the ground). In the model, parameter a is the horizontal distance from...
4. (14 points) For a linear 2-DOF model of a vehicle E(r) moving on a uneven road, (a) describe the base excitation y(t) when the vehicle is moving to the right at speed v; (b) derive equations o...
4. (14 points) For a linear 2-DOF model of a vehicle E(r) moving on a uneven road, (a) describe the base excitation y(t) when the vehicle is moving to the right at speed v; (b) derive equations of motion for the vehicle model; (b) build a Simulink model based on the equations of motion, using the blocks given below, with y() as the input and xi() and x2) as outputs. m2 x1(r) yt)input du/dt 1/s Derivative Integrator Sum Signal Generator...
5 Problem 4.41 Consider the second-order plant with transfer function: 1 G(s) = (s1)(5s 1) and...
5 Problem 4.41 Consider the second-order plant with transfer function: 1 G(s) = (s1)(5s 1) and in a unity feedback structure 1. Determine the system type and error constant with respect to tracking polynomial reference inputs of the system for P (i.e., De = kp), PD (i.e., Dc = kp+kps), and PID (i.e., De=kp +ki/s+ kps) controllers. Let kp = 19,ki 0.5,kp = 4/19. 2. Determine the system type and error constant of the system with respect to disturbance inputs...
Problem 5 Information
Problem 6: 10 points A) Find moments of first and second order and the variance for variable M in Problem 5 (B) Find expectation and variance for variable N in Problem 5. Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn} uniformly distributed over the unit interval, (0,1) Introduce two new random variables, M-max (Xi, X2,..., Xn) and N- min (X1, X2,... ,Xn)
Problem 1 Consider two first order low-pass systems connected in parallel: -2u The objective is to determine a second order ODE describing the variable y by manipulating the differential equations (no transfer function techniques are allowed). Answer the following series of questions: 1. (2 points) Write the variable y in terms of i and 2 2. (6 points) Determine the relationship between y, j, and z1, z2 and u. Write your final expression in a matrix-vector format: ? 01 ??...
0 2 10 0 2 8 Consider the multiple regression model where є¡ ~ iid Ņ(0, σ*) for i = i, 2, 3, 4, 5. (c) Fill in the values for the following ANOVA table: Source of Variation Sum of Squares df Mean Square F Regression on Xi, X2 Error Total (Corrected) (d) State the nul and alternative hypotheses associated with the F test from the ANOVA table in part (c) and do the F test (e) Compute R2 (f)...
Problem 2: Consider again the two RLC circuits from HW1 Problem 6 C L IKs IKs L In HW1 Problem 6 you found the transfer function Vc(s)V(s) for each of the circuits, using Impedance Analysis. You essentially assumed zero initial conditions (for the capacitor's voltage S and for the inductor's current) 2.1. Develop a state-variable model for each of the circuits, where the state variables are (in both circuits) xi vc and x2 i That is, derive (for each circuit)...
3. Consider the following production problem Maximize 10r 12r2 20r, subject to the constraints xi +x2 +x3 10. ri + 2r2 +3rs 3 22, 2x1 2a2 +4x3 S 30 120, x2 20, 0 (a) (2 points) Solve the problem using the simplex method. Hint: Check your final tableau very carefully as the next parts will depend on its correct- ness. You will end up having 1, 2, r3 as basic variables. (b) (6 points) For1,2, and 3, determine the admissible...
• (Problem 2) Consider the second version of the Lotka-Volterra model: F(a – bF – cS) dF dt ds dt S(-k + \F). (1) Explain the model; i.e. what are the terms in the equation signify? How is this model different from equations (1)? (2) Find the equilibrium point(s). (3) Linearize (2) about the equilibrium point(s). (4) Classify if the equilibrium points are stable or unstable. (5) Pick some values for a, b, c, k, 1. Plot the solutions of...
1. Consider the following model where y denotes the tool life and xi, x2, and tz denote the cutting speed, tool type, and type of cutting oil, respectively. There are two different tool types, A and B, and there are two type of cutting oils, low-viscosity oil and medium-viscosity oil. The two categorical predictors are 1 if type A 1 f low-viscosity oil used defined as *20 if type B and c3 0 if medium-viscosity oil used (a) Interpret the...
Problem 4 (10 points) Astrom et a1 have presented a simplified second-order transfer function model for bicycle dynamics as C1 Parameters defining the bicycle geometry. Schematic (a) top and (b) rear views of a bicycle The steer angle is δ and the roll angle is φ. The input is δ(t), the steering angle, and the output is Plt), the tilt angle (between the the bicycle longitudinal plane and the ground). In the model, parameter a is the horizontal distance from...
4. (14 points) For a linear 2-DOF model of a vehicle E(r) moving on a uneven road, (a) describe the base excitation y(t) when the vehicle is moving to the right at speed v; (b) derive equations of motion for the vehicle model; (b) build a Simulink model based on the equations of motion, using the blocks given below, with y() as the input and xi() and x2) as outputs. m2 x1(r) yt)input du/dt 1/s Derivative Integrator Sum Signal Generator...
5 Problem 4.41 Consider the second-order plant with transfer function: 1 G(s) = (s1)(5s 1) and in a unity feedback structure 1. Determine the system type and error constant with respect to tracking polynomial reference inputs of the system for P (i.e., De = kp), PD (i.e., Dc = kp+kps), and PID (i.e., De=kp +ki/s+ kps) controllers. Let kp = 19,ki 0.5,kp = 4/19. 2. Determine the system type and error constant of the system with respect to disturbance inputs...
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