Alternative form with detuning factor F

Quiz: SIMC PI-tunings SIMC-tunings QUIZ y y Step response t [s] Time t (a) The Figure shows the response (y) from a test where we made a step change in the input (Δu = 0.1) at t=0. Suggest PI-tunings for (1) τc=2,. (2) τc=10. (b) Do the same, given that the actual plant is

QUIZ Solution Actual plant:

Approximation of step response QUIZ Approximation of step response Approximation ”bye eye”

Tunings from Step response “by eye” model SIMC-tunings Kc=2.9, tauI=10 Kc=9.5, tauI=10 OUTPUT y INPUT u Tunings from Step response “by eye” model Setpoint change at t=0, input disturbance = 0.1 at t=50 Tunings from Half rule (Somewhat better) Kc=2, tauI=5.5 Kc=6, tauI=5.5

Half-rule approach: Approximation of zeros depends on tauc! QUIZ Half-rule approach: Approximation of zeros depends on tauc!

Some discussion points Selection of τc: some other issues Obtaining the model from step responses: How long should we run the experiment? Cascade control: Tuning Controllability implications of tuning rules

Selection of c: Other issues Input saturation. Problem. Input may “overshoot” if we “speedup” the response too much (here “speedup” = /c). Solution: To avoid input saturation, we must obey max “speedup”:

A little more on obtaining the model from step response experiments 1 ¼ 200 (may be neglected for c < 40) “Factor 5 rule”: Only dynamics within a factor 5 from “control time scale” (c) are important Integrating process (1 = 1) Time constant 1 is not important if it is much larger than the desired response time c. More precisely, may use 1 =1 for 1 > 5 c Delay-free process (=0) Delay  is not important if it is much smaller than the desired response time c. More precisely, may use  ¼ 0 for  < c/5 time  ¼ 1 (may be neglected for c > 5) c = desired response time

Step response experiment: How long do we need to wait? RULE: May stop at about 10 times effective delay FAST TUNING DESIRED (“tight control”, c = ): NORMALLY NO NEED TO RUN THE STEP EXPERIMENT FOR LONGER THAN ABOUT 10 TIMES THE EFFECTIVE DELAY () EXCEPTION: LET IT RUN A LITTLE LONGER IF YOU SEE THAT IT IS ALMOST SETTLING (TO GET 1 RIGHT) SIMC RULE: I = min (1, 4(c+)) with c =  for tight control SLOW TUNING DESIRED (“smooth control”, c > ): HERE YOU MAY WANT TO WAIT LONGER TO GET 1 RIGHT BECAUSE IT MAY AFFECT THE INTEGRAL TIME BUT THEN ON THE OTHER HAND, GETTING THE RIGHT INTEGRAL TIME IS NOT ESSENTIAL FOR SLOW TUNING SO ALSO HERE YOU MAY STOP AT 10 TIMES THE EFFECTIVE DELAY ()

“Integrating process” (c < 0.2 1): Need only two parameters: k’ and  From step response: Example. Step change in u: u = 0.1 Initial value for y: y(0) = 2.19 Observed delay:  = 2.5 min At T=10 min: y(T)=2.62 Initial slope: Response on stage 70 to step in L y(t) 2.62-2.19 7.5 min =2.5 t [min]

Example (from quiz) OUTPUT y INPUT y tauc=10 tauc=2 Step response Δu=0.1 Assume integrating process, theta=1.5; k’ = 0.03/(0.1*11.5)=0.026 SIMC-tunings tauc=2: Kc=11, tauI=14 (OK) SIMC-tunings tauc=10: Kc=3.3, tauI = 46 (too long because process is not actually integrating on this time scale!) OUTPUT y INPUT y tauc=10 tauc=2

Cascade control Cascade control

Tuning of cascade controllers • Want to control y (primary CV), but have “extra” measurement y 1 2 • Idea: Secondary variable (y ) may be tightly controlled and this 2 helps control of y . 1 • Implemented using cascade control: Input (MV) of “primary” controller (1) is setpoint (SP) for “secondary” controller (2) • Tuning simple: Start with inner secondary loops (fast) and move upwards • Must usually identify ”new” model ( G1’ = G1 G21 K2 (I+K2G22)-1 ) experimentally after closing each loop • One exception: Serial process, G21 = G22 2 – Inner (secondary - 2) loop may be modelled with gain=1 and effective delay=( t + q ) c 2 See next slide

Special case: Serial cascade Cascade control Special case: Serial cascade y2 = T2 r2 + S2d2, T2 = G2K2(I+G2K2)-1 K2 is designed based on G2 (which has effective delay 2) then y2 = T2 r2 + S2 d2 where S2 ¼ 0 and T2 ¼1 · e-(2+c2)s T2: gain = 1 and effective delay = 2+c2 NOTE: If delay is in meas. of y2 (and not in G2) then T2 ¼ 1 ·e-c2s SIMC-rule: c2 ≥ 2 Time scale separation: c2 ≤ c1/5 (approximately) K1 is designed based on G1’ = G1T2 same as G1 but with an additional delay 2+c2

Example: Cascade control serial process G1 u y1 K1 ys G2 K2 y2 y2s Use SIMC-rules! Without cascade With cascade

Tuning cascade control

Tuning cascade control: serial process Inner fast (secondary) loop: P or PI-control Local disturbance rejection Much smaller effective delay (0.2 s) Outer slower primary loop: Reduced effective delay (2 s instead of 6 s) Time scale separation Inner loop can be modelled as gain=1 + 2*effective delay (0.4s) Very effective for control of large-scale systems

Alternative closed-loop approach: Setpoint overshoot method Procedure: Switch to P-only mode and make setpoint change Adjust controller gain to get overshoot about 0.30 (30%) Record “key parameters”: 1. Controller gain Kc0 2. Overshoot = (Δyp-Δy∞)/Δy∞ 3. Time to reach peak (overshoot), tp 4. Steady state change, b = Δy∞/Δys. Estimate of Δy∞ without waiting to settle: Δy∞ = 0.45(Δyp + Δyu) Advantages compared to Ziegler-Nichols: * Not at limit to instability * Works on a simple second-order process. Closed-loop step setpoint response with P-only control. M. Shamsuzzoha and S. Skogestad, ``The setpoint overshoot method: A simple and fast method for closed-loop PID tuning'', Journal of Process Control, 20, xxx-xxx (2010)

Proposed PI settings (including detuning factor F) Setpoint overshoot method Summary setpoint overshoot method From P-control setpoint experiment record “key parameters”: 1. Controller gain Kc0 2. Overshoot = (Δyp-Δy∞)/Δy∞ 3. Time to reach peak (overshoot), tp 4. Steady state change, b = Δy∞/Δys Proposed PI settings (including detuning factor F) Choice of detuning factor F: F=1. Good tradeoff between “fast and robust” (SIMC with τc=θ) F>1: Smoother control with more robustness F<1 to speed up the closed-loop response.

Example: High-order process Setpoint overshoot method Example: High-order process P-setpoint experiments Closed-loop PI response

Example: Unstable plant Setpoint overshoot method Example: Unstable plant First-order unstable process No SIMC settings available Closed-loop PI response

A comment on Controllability (Input-Output) “Controllability” is the ability to achieve acceptable control performance (with any controller) “Controllability” is a property of the process itself Analyze controllability by looking at model G(s) What limits controllability?

Controllability Recall SIMC tuning rules 1. Tight control: Select c= corresponding to 2. Smooth control. Select Kc ¸ Must require Kc,max > Kc.min for controllability ) max. output deviation initial effect of “input” disturbance y reaches k’ ¢ |d0|¢ t after time t y reaches ymax after t= |ymax|/ k’ ¢ |d0|

CONTROLLABILITY Controllability

Example: Distillation column CONTROLLABILITY Example: Distillation column

Example: Distillation column CONTROLLABILITY Example: Distillation column

Conclusion controllability If the plant is not controllable then improved tuning will not help Alternatives Change the process design to make it more controllable Better “self-regulation” with respect to disturbances, e.g. insulate your house to make y=Tin less sensitive to d=Tout. Give up some of your performance requirements