1. Process Control Basics Distillation Control Course December 4-8, 2004

2. Transfer Functions SISO (Single Input Single Output) Steady-state “Increasing L from 1.0 to 1.1 changes y D from 0.95 to 0.97” (for example, run Hysys) Dynamics “The response has a dead time of 2 minutes and then rises with a time constant of 50 minutes” time yDyD 10 0.97 63% 0.96 0.95 Θ= 2 50 τ=50 Laplace: Transfer function: Linear process: Gain independent of input magnitude (  L)

3. Alternative Method of Obtaining Transfer Function: Take laplace of linearized model Note: Gain is obtained by using s = 0: “gain” = g(0) “gain”=1 Time constant τ=Mi/L Example L (constant) x i+1 M i =constant LxiLxi Response: X i+1 XiXi time τ

4. MIMO (multivariable case) Direct Generalization: “Increasing L from 1.0 to 1.1 changes y D from 0.95 to 0.97, and x B from 0.02 to 0.03” “Increasing V from 1.5 to 1.6 changes y D from 0.95 to 0.94, and x B from 0.02 to 0.01” Steady-State Gain Matrix (Time constant 50 min for y D ) (time constant 40 min for x B )

5. Important Advantages With Transfer Matrices 1. G(s) is independent of input u! For given u(s) compute output y(s) as Can therefore make block diagrams G(s) u(s) y(s) 2. Frequency Response: G(s), with s=j  (pure complex no.) gives directly steady-state response to input sin(  t)! g 11 (s) u1u1 y1y1

6. “Steady-state” frequency response Response (as things settle) time y 1 (t) u 1 (t) A A ¢ |g 11 (jw)| P

8. Note Tells directly how much a sine of frequency w is amplified by process w (log scale) Fast sinusoids are “filtered by process (don’t come through BODE PLOT (magnitude only) 0.1 1 2 |g 11 | |g 11 (jw)| (log scale)| 0.010.020.1 =1/50

9. Effect of Feedback Control C(s) ysys u G(s) d y G(s): process (distillation column) C(s): controller (multivariable or single-loop PI’s) y: output, y s : setpoint for output u: input d: effect of disturbance on output

10. Negative feedback With feedback control Eliminate u(s) in (1) and (2). Set y s =0 S (S) =“sensitivity function” = surpression disturbances

11. In practice: Cannot do anything with fast changes, that is, S (jw) = I at high frequency S is often used as performance measure Log- scale 1 0.1 Small resonance peak (small peak = large GM and PM) Bandwidth (Feedback no help) Slow disturbance d 1 : 90% effect on y 2 removed w (log) w B τ Bandwidth frequency  B ¼ 1/  c  c [s] = closed-loop response time