Did you ever find the answer for this??? I know there are a couple of different mathematical methods for calculating (tuning) PID control loops... One of the most common is the Ziegler-Nichols Method. So I'll use that method as the example. I know this loop tuning class is coming a little too late for your final, but maybe this will help someone else in your situation.... (Its not letting me post my math formula pics on here..I'll work on that so this will be a little more understandable...
~April~
A little background on control loop theory for the class....
The process stability of a PID control loop depends upon the proportional, integral and derivative constants used. Using the conventional tangent method and the proper tuning rule, the optimum P, I and D can be estimated. With this optimum P, I and D set into the controller, an optimum response is normally achieved.
Currently, there are several equations built into control system PID’s control algorithms that are capable of stabilizing a control loop’s response.
Despite the variation in the equations used for the algorithm, the variables used remain the same i.e. proportional constant (P) or controller gain (Kc), integral constant (I), derivative constant (D), set-point value (SP) and measurement value (PV). In fact, the units of I and D may differ from one instrument manufacturer to another instrument manufacturer. For instance, the I constant, one manufacturer may use min per repeat (integral time) while other manufacturer may use repeat per min (integral gain) [1]. Integral time and integral gain is inversely related to each other.
The variation in these control algorithms of PID controllers only affect the shape and size, but not the characteristics, of the process response curve. These characteristics of PID controller are the tendency to produce overshoot, undershoot, off-set and oscillation in the system response.
The selection of P, I, and D values is very crucial. They determine whether the process is oscillatory, stable or unstable. To obtain a stable process, numerous combinations of P, I and D values are possible, but there is only one combination that will produce an optimum response curve.
One quick method in getting the optimum P, I and D is by using the conventional tangent method [2]. It provides two most vital information about the process dynamic i.e. the deadtime and the response rate. This information is used in the tuning rules, such as Zieglar-Nichols, to estimate the optimum P, I and D for the controller. Utilizing trend data are the common way to record the process response curve.
III. TANGENT METHOD & OPTIMUM PID
The tangent method starts with an openloop test. It is done by putting the controller in manual mode and making a load change (DMV) of 5 to 20% to the controller’s output. The resulted response curve is recorded until a new steady state level has been reached or until an ample amount of data is obtained necessary to perform the analysis. The response curve is then analyzed for the process deadtime (Td) and the response rate (RR) by drawing a tangent line to the steepest point of the response curve. By definition, the process deadtime is estimated at the cross section between the baseline of the old steady-state level and the tangent line [2,3]. Figure 1 shows the load change made (DMV), the drawn tangent line and the estimated process deadtime (Td).
Figure 1: A step change of DMV (bottom) and the associated response curve (top).
Td (deadtime) and RR (response rate) are incorporated in the tuning rule for the optimum PID calculation. There are six openloop tuning rules, which has been compiled by Senbon and Hanabuchi [5]. One of the famous openloop tuning rule is Zieglar-Nichols as shown in Table 1 below.
Ziegler-Nichols Method:
First, note whether the required proportional control gain is positive or negative. To do so, step the input u up (increased) a little, under manual control, to see if the resulting steady state value of the process output has also moved up (increased). If so, then the steady-state process gain is positive and the required Proportional control gain, Kc, has to be positive as well.
Turn the controller to P-only mode, i.e. turn both the Integral and Derivative modes off.
Turn the controller gain, Kc, up slowly (more positive if Kc was decided to be so in step 1, otherwise more negative if Kc was found to be negative in step 1) and observe the output response. Note that this requires changing Kc in step increments and waiting for a steady state in the output, before another change in Kc is implemented.
§When a value of Kc results in a sustained periodic oscillation in the output (or close to it), mark this critical value of Kc as Ku, the ultimate gain. Also, measure the period of oscillation, Pu, referred to as the ultimate period. ( Hint: for the system A in the PID simulator, Ku should be around 0.7 and 0.8 )
Using the values of the ultimate gain, Ku, and the ultimate period, Pu, Ziegler and Nichols prescribes the following values for Kc, tI and tD, depending on which type of controller is desired:
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What is the controller gain for a damping ratio of 0.30? 
