## PRACTICE PROBLEMS

1. Convert the following functions from time to laplace functions using the tables. 2. Convert the following functions below from the laplace to time domains using the tables. 3. Convert the following functions below from the laplace to time domains using partial fractions and the tables. 4. Convert the output function below Y(s) to the time domain Y(t) using the tables. 5. Convert the following differential equations to transfer functions. 6. Given the transfer function, G(s), determine the time response output Y(t) to a step input X(t). 7. Given the following input functions and transfer functions, find the response in time. 8. Do the following conversions as indicated. 9. Convert the output function to functions of time. 10. Solve the differential equation using Laplace transforms. Assume the system starts undeflected and at rest. 11. For the following control system select a controller transfer function, Gc, that will make the overall system performance match the desired transfer function. 12. Write a C program for an ATMega microcontroller to implement the control system in the dashed line below with an update time of 10ms. 13. A feedback control system is shown below. The system incorporates a PID controller. The closed loop transfer function is given. a) Develop the transfer function for the system.

b) Select controller values that will result in a response that includes a natural frequency of 2 rad/sec and damping factor of 0.5. Verify that the controller will be stable.

c) If the values of Kp = Ki = Kd = 1 find the response to an input of 5sin(10t) as a function of time using the LaPlace Transforms.

d) Find the response in part c) using numerical methods. Show the results as a table and graph. The results should show the region(s) of greatest interest.

e) Find the system response to an input of X = 5sin(100t + 1) using phasor transforms.

14. a) The block diagram below is for an angular positioning system. The setpoint is a desired angle, which is converted to a desired voltage. This is compared to a feedback voltage from a potentiometer. A PID controller is used to generate an output voltage to drive a DC motor. Simplify the block diagram. b) Given the transfer function below, select values for Kp, Ki and Kd that will result in a second order response that has a damping factor of 0.125 and a natural frequency of 10rad/s. (Hint: eliminate Ki). c) The function below has a step input of magnitude 1.0. Find the output as a function of time using numerical methods. Give the results in a table OR graph. d) The function below has a step input of magnitude 1. Find the output as a function of time by integrating the differential equation (i.e., using the homogeneous and particular solutions). e) The function below has a step input of magnitude 1. Find the output as a function of time using Laplace transforms. f) Given the transfer function below; 1) apply a phasor transform and express the gain and phase angle as a function of frequency, 2) calculate a set of values and present them in a table, 3) use the values calculated in step 2) to develop a frequency response plot on semi-log paper, 4) draw a straight line approximation of the Bode plot on semi-log paper. 15. a) Simplify the block diagram as far as possible. b) Given the transfer function below, select values for Kp and tau that will include a second order response that has a damping factor of 0.125 and a natural frequency of 10rad/s. c) The function below has a step input of magnitude 1.0. Find the output as a function of time using numerical methods. Give the results in a table OR graph from 0.0 to 0.010s. d) The function below has a step input of magnitude 1. Find the output as a function of time by integrating the differential equation (i.e., using the homogeneous and particular solutions). e) The function below has a step input of magnitude 1. Find the output as a function of time using Laplace transforms. f) Given the transfer function below; 1) apply a phasor/Fourier transform and express the gain and phase angle as a function of frequency, 2) calculate a set of values and present them in a table, 3) use the values calculated in step 4) to develop a frequency response plot on semi-log paper, 5) draw a straight line approximation of the Bode plot on semi-log paper. g) Select a controller transfer function, Gc, that will reduce the system to a first order system with a time constant of 0.5s, as shown below. 16. a) Find the simplified transfer function for the block diagram. b) Given the transfer function below calculate a controller transfer function, Gc. The desired response should be first order with a time constant of 1s. c) For the system given below, and the provided controller function, find the response to a unit step input using LaPlace transforms. d) For the system given below, and the provided controller function, find the response to a unit step input by solving the differential equation. e) For the system given below, and the provided controller function, find the response to a unit step input using numerical methods. f) Write a C program for an ATMega microcontroller to implement the control system in the dashed line below with an update time of 10ms. 17. A feedback control system is shown below. a) Develop the transfer function for the system.

b) Select controller values that will result in a response that includes a natural frequency of 10 rad/sec and damping factor of 2.0.

18. a) Using Laplace transforms find the system response for the given transfer function and input function. b) Convert the transfer function to a differential equation and then solve the differential equation with the given input. c) Convert the given transfer function to state equations and then solve it numerically. Show the results for the first 2 seconds in 0.2s intervals. d) Compare the solutions for parts a), b), and c). Show that they are equal.

19. a) Given the experimental Bode (Frequency Response Function) plot below, find a transfer function to model a positioning system. The input is a voltage ‘V’ and the output is a displacement ‘x’. b) The transfer function found in step a) will be used for the positioning system in the block diagram below. Find the overall transfer function for the system. 20. Write a C subroutine for an ATMega microcontroller to implement the control system in the dashed line below with an update time of 10ms. The subroutine should use integer math for all calculations 