1. 1 Effect of Input Rate Limitation on Controllability Presented at AIChE Annual Meeting in Austin, Texas November 7 th, 2002 Espen Storkaas & Sigurd Skogestad Presented by Federico Zenith Department of Chemical Engineering, Norwegian Unviversity of Science and Technology

2. 2 Anti Slug Control Unstable flow in multiphase pipeline-riser systems for offshore oil production stabilized by pressure controller How fast must be the topside choke valve (actuator) be? –Slow for safety reasons –Fast for control Wells PT PC Pressure set point Measured pressure at riserbase Topside Choke valve Subsea choke

3. 3 Outline Motivation Controllability analysis with input rate limitation Required input rates for: –Stabilization –Disturbance rejection and set point tracking Controller design with input rate limitations

4. 4 Controllabilty analysis Definition (Skogestad & Postlethwaite (1996)) (Input-output) controllability is the ability to achieve acceptable control performance; that is, to keep the outputs (y) within specified bounds or displacements form their references (r), in spite of unknown but bounded variations, such as disturbances (d) and plant changes, using available inputs (u) and available measurements (y m or d m ). Controllability is independent of the controller, and is a property of the plant or process itself

5. 5 Effect of input limitations Open-loop unstable systems: –Input saturation breaks feedback loop –Stabilizing effect ruined –Critical Open-loop stable systems: –Input saturation limits performance Good theoretical foundation for magnitude saturation Here: Extend theory to cover rate limitations on actuator Input rate limitation du/dt · –Actuator from u=-1 to u=1 in 2/ sec u (from controller) -1 · u · 1 Magnitude limitation du/dt · Rate limitation u (to process)

6. 6 Effect of input rate limitations Sinusoids with frequency  through rate limiter, rate limited to be less than :   /2  Dashed lines are input (controller output), solid lines are outputs from rate limiter

7. 7 Bound on available input Maximal slope for u ( t )=| u ( j )| ¢ sin (t) at t=0: Rate limitation du / dt ·, | u ( j )| · / Magnitude limitation | u ( j )| · 1 Alternative: –|u|<1 (magnitude) –

8. 8 Input rates for stabilizing control For any unstable pole p (Havre & Skogestad, 2001), Magnitude saturation (|u| · 1): Rate saturation (|su(s)| · ): u G(s) G d (s) d · 1 + +y K(s)

9. 9 Example 1 – Stabilization Perfect disturbance rejection: – OK! – fsda u Rate Limiter du / dt · Magnitude Limiter d + + y

10. 10 Input rates for perfect and acceptable control Perfect control: | u |= | G -1 G d | ¢ | d | Input limitation requires – (magnitude limitation) – (rate limitation) Acceptable control (| e |=1): | u |=| G -1 |(| G d | ¢ | x |-1) – (magnitude limitation) – (rate limitation) Set point tracking: R instead of G d

11. 11 Example 2 – Disturbance Rejection Perfect disturbance rejection: – OK! – fsda Acceptable disturbance rejection: ¸ 0.12 sec -1 u Rate Limiter du / dt · Magnitude Limiter d + + y

12. 12 Achievable bound on input Exact bound has infinite- dimensional representation in frequency domain Cannot be used in direct controller design Bound approximated by 1 st order lag filter:

13. 13 –New limit for example 1: ¸ 0.4 sec -1 (0.22 sec -1 with exact bound) Required input rates using approximate bound on input Perfect disturbance rejection: –New limit for example 2: ¸ 1.02 sec -1 (1 sec -1 with exact bound) Stabilization:

14. 14 Controller design H 1 design with minimum input rate (minimize W u KSG d ): u d + +y K(s) p

15. 15 Simulation Disturbance with magnitude 1 and frequency p:

16. 16 Conclusions Limitation on input rate can limit performance for control systems Explicit lower bounds on required input rate derived –Stabilization –Disturbance rejection and set point tracking Controller design with limited input rates