Summary of Week 3
    1. Mulitiple Linear ODE System (review and discussion of derivation of determinant formula)
    2. Some Typical Process Dynamics
      1. First Order System
        Standard Form: t (dx/dt ) + x = f(t)

        Characteristics: t = time constant

        If f(t) = Kp = constant,
          x( t ) = Kp - [ Kp-x(0) ] exp( -t/t )
          [ x( t ) - x(0) ] = 0.632 [ Kp - x(0) ]

      1. Second Order System
Standard Form:
Characteristic equation: tn2 s2 + 2tn z s+1 = 0


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.
    1. Stability Conditions for Second Order Systems
      From the characteristic equation of a second order system,
      the process is stable if and only if all the coefficients have the same sign.
    3. Exam during 12/16/99. Reviewed Solution 12/17/99
    1. Introduction to PID Control


      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