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

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.  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
2.
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

2.

 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