5 June 2012 Dept. of ECE Drexel University

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ECE-S513 Fundamentals of Systems III (Moshe Kam): Final Exam

Instructions: This exam covers nonlinear feedback control design tools as presented in Khalil Chapter 12, pages 469-485, particularly the topics of stabilization via linearization, output feedback stabilization via linearization, and integral control via linearization. This exam must be completed individually. Please submit the following:

1. A report containing answers to all questions, including derivations, figures, and supporting explanations.

2. Appropriately documented MATLAB code used to answer questions. This code should include a script named final_exam.m that performs the necessary computations and outputs the requested results and figures.

Problem Statement Consider the forced pendulum equation:

= sin() + Where a, b and c are constants given by:

=

, =

, = 12

Where is the acceleration due to gravity, is the length of the rod, is the mass of the bob, is the angle subtended by the rod and the vertical axis through the pivot, and is the coefficient of friction. is the input torque applied to the pendulum.

The goal is to design an input to stabilize the pendulum at an arbitrary angle = . Choosing state variables 1 = , 2 = and a control variable = where = sin() is the steady state torque. The resulting state equation is given by:

1 = 2 2 = (sin(1 + ) sin()) 2 + = 1

Use the above state equation representation of the system to answer the following questions.

5 June 2012 Dept. of ECE Drexel University

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Preliminary Calculations

Question 1: Verify the provided state equation by performing the appropriate substitutions in the original forced pendulum equation.

Question 2: Assume the input torque = 0. Solve for the equilibrium points of the system. At what angle does the pendulum hang at equilibrium?

Question 3: Linearize the system about the origin. Prove that the linearized system is controllable. Taking = [1 2], prove that ( ) is Hurwitz for:

1 > cos , 2 = Question 4: Prove that the linearized system is observable. Taking = [1 2] , prove that ( ) is Hurwitz for:

1 + > 0, 1 + 2 + cos() > 0 Stabilization via linearization

For the remaining questions, consider the following values:

= 10, = 0.1, = 1, = 3 Question 5: Write a function system(t,x) that contains the state-variable description of the system when the input torque = 0. Use ODE45 to simulate the response of the system with initial conditions 1 = 2 , 2 = 0. Plot the angle of the pendulum vs. time over a representative time span. Verify that this result confirms your result from Question 2.

Question 6: Design state feedback gains such that the eigenvalues of ( ) are {2,2} using any method of your choosing. Write a function system_state_feedback(t,x) that contains the state variable description of the system with full state feedback. If you also wish to pass your gains to this function, read the help for ODE45. Use ODE45 to simulate the response of the system with initial conditions 1 = 2 , 2 = 0. Plot vs. time over a representative time span. Plot the torque vs. time over the same time span.

Question 7: Design observer gains such that the eigenvalues of ( ) for the state observer = + +( 1) are {-10,-10}. Write a function system_output_feedback(t,x) that contains the state variable description of the system with output feedback. Use ODE45 to simulate the response of the system with initial conditions 1 = 2 , 2 = 0 and initial conditions for the observer states of [1 2] + where = 0.1. Overlay a plot of vs. time over the plot of vs. time for full state feedback. Overlay a plot the torque vs. time over the plot of vs. time for full state feedback. Compare the results using output feedback and full state feedback. What happens as you increase ?

5 June 2012 Dept. of ECE Drexel University

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Integral control via linearization For integral control, we do not require that we apply the steady state torque in order to apply our control law. Instead, we consider = , which gives rise to the state equation

1 = 2 2 = sin(1 + ) 2 + = 1

Question 8: Following the methods put forth in section 12.4 in Khalil, form the matrices ,, using 1 =[1 2], and 2 = [3]. Prove that is Hurwitz if: + 2 > 0, ( + 2)( cos() + 1) 3 > 0, 3 > 0

Recall:

= 0 0 , = 0 , = [1 2]

Question 9: Design state feedback gains such that the eigenvalues of ( ) are {2,2,4} using any method of your choosing. Write a function system_pid(t,x) that contains the state variable description of the system using the control law = 1( ) 2 3 where = . Use ODE45 to simulate the response of the system with initial conditions 1 = 2 , 2 = 0 and = 0. Plot vs. time over a representative time span, overlaid on top of the previous two plots. Plot the torque vs. time over the same time span, overlaid on top of the previous two plots. Compare the results of each of the three controllers.

Control under parameter perturbations

Question 10: Assume that the mass of the bob is perturbed, but your control rules and parameters are unchanged. Try increasing the mass by 10%, 20%, etc. Plot vs. time and vs. time for the state feedback control, output feedback control, and integral control rules. Plot vs. time for the original parameters and perturbed parameters on the same set of axes. Plot the torque vs. time for the original parameters and perturbed parameters on the same set of axes. Provide the plot when mass is increased 10% with your report. How to the controllers compare under this perturbation?

Compare the results obtained with each controller. Which controller(s) fail first as you increase the mass? How far can you increase the mass before each controller fails? Compute the theoretical point of failure of the integral controller using the result presented at the bottom of page 483. How does your result compare?

Question 11: In the previous questions, we designed a feedback control law in the form of equation 12.20 in Khalil where 3 = 0. What does the term 3 control? Why were we able to set it equal to zero? Redo question 9 where 3 is nonzero. Plot vs. time and vs. time for both the case where 3 = 0 and 3 0 on the same two sets of axes. Compare the performance of the different control laws. Include in your plot the results for a few different interesting values for 3. How does 3 affect the controller performance?

Problem StatementPreliminary CalculationsStabilization via linearizationIntegral control via linearizationControl under parameter perturbations