The output of this block is the heading angle, psi, and the input to the block is the actual rudder angle, delta_r. In the absence of a disturbance, the actual and commanded rudder angles are equal.

     TASKS:

1. Design a compensator G_c(s) such that the following specifications are all satisfied:

   • when the reference input signal is 0, and the disturbance signal is a step function, the steady-state value of the actual heading angle must be 0;
   • when the reference input signal is a step function, and the disturbance signal is 0, the settling time of the actual heading angle must not exceed 200 seconds;
   • when the reference input signal is a step function, and the disturbance signal is 0, the actual heading angle must not exceed the desired heading angle by more than 20%;
   • when the reference input signal is a step function equal to 10 degrees, and the disturbance signal is 0, the commanded control signal (rudder angle) must not exceed 15 degrees in absolute value.

2. Perform simulations in MATLAB with your compensator design to verify that all the specifications have been satisfied. Plots should be made that indicate the responses of the various signals are satisfactory. Determine the steady-state error in heading angle when the reference input is a unit ramp signal and the disturbance signal is zero.
   
   It should be noted that there is not a settling time requirement for a step input at the disturbance signal, so that response may take longer than the other responses. Ramp responses may also be longer than step responses.
3. This part of the project deals with the stability robustness of your design, that is, how much change can there be in the plant and/or controller models without losing closed-loop stability? My MATLAB function tsypkin.m can be used for this. The zip file containing this function (and my MATLAB function circle.m) should be downloaded and unzipped. (If you have trouble downloading the file, email me, and I will email the 2 functions back to you). The two functions should be put into your working MATLAB directory. For your purpose in this project, the syntax for using the function can be
   
   rho = tsypkin(dcl, 0.15*dcl);

   where dcl is the closed-loop characteristic polynomial of your final compensated system. The function will create a figure with 3 plots on it, and will return a 3-element vector rho. A sample plot is shown for illustration. The elements in the rho vector are the maximum allowed perturbations to the coefficients of the closed-loop characteristic equation in terms of the 1-norm, 2-norm, and infinity-norm, respectively. The larger the values in rho, the more robust your control system is to changes in these coefficients. Evaluate your design using this function. It is hoped that all the values in rho will be greater than 1.

     REPORT:

    You must provide a typed report that documents your design activity and provides plots that verify that all the specifications are satisfied. Handwritten reports will not be accepted. If you wish, you may produce your reports, including plots, in the form of web pages rather than in hardcopy form. In that case, you must provide me with the URL for your web pages by the time that the project report is due.

    Your report must describe your design approach and explain how the approach is able to satisfy the design specifications. Your report must reference the plots that are included in it. The plots must clearly demonstrate that the specifications are satisfied, and the plots must be clearly labeled with axis labels and titles.

    Include in your report a discussion on why robustness of the control system should be included in your design process, or at least evaluated when the design is complete.