Summary of Week 3

1. Some special parametric forms of second order systems:

Second order systems:  tn2 d2x/dt2 + 2 z tn dx/dt + x = f(t)
where  1/tn = natural frequency
            z = damping coefficient
Eigenvalues:
Case 1: z 1 (overdamped) then s1 and s2 are both real and negative.
Case 2: z=1 (critical damping) then s1 and s2 are equal.
Case 3: 0<z<1 (underdamped) then s1 and s2 are conjugate pair of complex roots,
Case 4: z=0 (undamped) then s1 and s2 are both pure imaginary roots.
Case 5: z<0 (unstable) then s1 and s2 are complex pair with positive real parts.
Solution to underdamped with f(t)=Kp, x(0)=xo and dx/dt(0) = 0.
with the following additional characteristics:
where x1* is the value of the first peak and x2* is the value of the second peak overshooting beyond Kp.
2. Linearization:
Objective:
To provide a linear approximation of a process model that could predict local behavior around a chosen operating condition.
Procedure:
Given:
                                        (1)
To obtain a linearized model of equation (1) with respect to the point: xo, uo and do, first obtain the following constants:
then the linearized model is given by
3. Perturbation Variables:  xp = x - xss

4. General elements of Feedback Control

5. Introduction to PID Control
 

      Type
      Algorithm
      Features
      Caution
      Proportional Control
      u = ubias + kc e
      Simple. Only one tuning parameter
      Offset present
      Proportional Integral Contol
      u = ubias +kc [e + (1/tI) ò e dt ]
      Removes offset. Can decrease response times.
      Can introduce low damping coefficient.
      Proportional Integral Derivative Contol
      u = ubias+kc [ e + (1/tI) ò e dt + tD de/dt
      Can reduce overshoot.
      Derivative term needs accompanying filter.
Where e = error = xset - x , kc = controller gain , tI = integral time , tD = derivative time