Summary of Week 5

1.PID Controllers

          Quick rules for selection:

             a.Offset is acceptable ; use P-control
             b.Fast process, offset undesirable: use PI-control
             c.Slow process, offset undesirable: use PID-control

2.Controller Tuning Methods

                  a. Ziegler-Nichols Method

                        i.Set controller to Proportional only control mode
                       ii.Obtain the ultimate gain, Ku, and the corresponding ultimate period, Pu, by slowly increasing
                         process gain until process sustains periodic oscillation
                       iii.Using Ku and Pu, evaluate the controller parameters using the Ziegler-Nichols tuning table.

                  b. Autotuning Relay Method

                        i.Connect the feedback to an on-off controller:

                         If process gain is positive: U = uo+h (if y<yset), uo-h (otherwise)

                         If process gain is negative: U = uo-h (if yyset) uo+h (otherwise)

                       ii.Wait until the process attains a limit cycle
                       iii.Using the amplitude of the limit cycle, a, evaluate the approximate ultimate gain, Ku =
                         (4/p)(h/a). Also record the period of oscillation of the limit cycle as the ultimate period, Pu.
                       iv.Using the calculated Ku and Pu, use the Ziegler-Nichols tuning table.
 

                c. Cohen and Coon Method:

      i) Under manual control, generate a process reaction curve by introducing a step change in the manipulated variable.
      ii)From the reaction curve, determine the process gain, K, the time constant, t, and the apparent time delay, tdelay. Using these values, determine prescribed controller parameters.
3. Laplace Transform:
Motivation: to transform a function in time into a corresponding function in Laplace variable s, with the expectation that in the new domain, the usual algebraic operations such as multiplication, division, addition and subtraction will make analysis and design of dynamic systems simpler.


Procedure:

      i) Given f(t), multiply this first with exp(-st).
      ii) Then integrate this product with respect to time from t = 0 to t = infinity;.
      iii) Simplify the integral, e.g. using integration by parts, etc., until you obtain a function of s.
(Note: s is a complex variable having a positive real part.)
 
Tables of Useful Functions:
Function Description
f (t)
L[f (t)]
Step Function
m(t)
1/s
Impulse Function
d(t)
1
Ramp Function
t
1/s2
Sine Function
sin(wt)
w/(s2+w2)
Cosine Function
cos(wt)
s/(s2+w2)
Exponential Function
exp(-a t)
1/(s+a)
Power Function
tN
N!/sN+1