Advances in PID Control

Transcription

1 Advances in PID Control Karl Johan Åström Department of Automatic Control, Lund University March 1, 218

2 Introduction Awareness of PID and need for automatic tuning: KJs Telemetric Experience 79-8, Westrenius Euroterm and Mike Sommerville The idea - Automatic generation of good input signals The patents Tore+KJ: Sweden 83, USA 85,... Commercial exploitation NAF Control, Sune Larsson, Ideon: Tore, Båth, SDM 2 (84), ECA 4 (86) Ahlsell, Alfa Laval Automation, Satt Control, ABB Fisher Controls, Advisory board 88-92, board of directors Fisher Controls + Rosemount Emerson Product development - From 2kbytes to Mbytes Gain scheduling, continuous adaptation Research: papers, MS, Lic & PhD Research Goals Understand PID control and its use How good models are required? How to find tuning rules - computation dependent This lecture: What have we learned?

3 Tore 4 Years of Collaboration Phd student 1978, PhD 1983; New Estimation Techniques for Adaptive Control Relay auto-tuning - patent 1983 NAF Back to the department at LTH 1999 PID control

4 Recent PhD Students Kristian Soltesz 213 On automation in Anesthesia Vanessa Romero 214 CPU Resource Management and Noise Filtering for PID Control Olof Garpinger 215 Analysis and Design of Software-Based Optimal PID Controllers Martin Hast 215 Design of Low-Order Controllers using Optimization Techniques Josefin Berner 217 Automatic Controller Tuning using Relay-based Model Identification Fredrik Bagge Carlson 21X Side projects: Optimization Julia programming

5 The Magic of Feedback Feedback has some amazing properties, it can make good systems from bad components, make a system insensitive to disturbances and component variations, stabilize an unstable system, create desired behavior, for example linear behavior from nonlinear components. The major drawbacks are that feedback can cause instabilities sensor noise is fed into the system PID control is a simple way to enjoy the Magic!

6 The Amazing Property of Integral Action Consider a PI controller u = ke+k i t e(τ)dτ Assume that all signals converge to constant values e(t) e, u(t) u and that t (e(τ) e )dτ converges, then e must be zero. Proof: Assume e =, then t t ( ) u(t) = ke + k i e(τ)dτ = ke + k i e(τ) e dτ + ki e t The left hand side converges to a constant and the left hand side does not converge to a constant unless e =, furthermore ( ) u( ) = k i e(τ) e dτ A controller with integral action will always give the correct steady state provided that a steady state exists. Sometimes expressed as it adapts to changing disturbances.

7 Predictions about PID Control 1982: The ASEA Novatune Team 1982 (Novatune is a useful general digital control law with adaptation): PID Control will soon be obsolete 1989: Conference on Model Predictive Control: Using a PI controller is like driving a car only looking at the rear view mirror: It will soon be replaced by Model Predictive Control. 22: Desborough and Miller (Honeywell): Based on a survey of over 11 controllers in the refining, chemicals and pulp and paper industries, 98% of regulatory controllers utilise PID feedback. The importance of PID controllers has not decreased with the adoption of advanced control, because advanced controllers act by changing the setpoints of PID controllers in a lower regulatory layer.the performance of the system depends critically on the behavior of the PID controllers. 216: Sun Li A recent investigation of 1 boiler-turbine units in the Guangdong Province in China showed 94.4% PI, 3.7% PID and 1.9% advanced controllers

8 Entech Experience & Protuner Experiences Bill Bialkowsk Entech - Canadian consulting company for pulp and paper industry Average paper mill has 3-5 loops, 97% use PI the remaining 3% are PID, MPC, adaptive etc. 5% works well, 25% ineffective, 25% dysfunctional Major reasons why they don t work well Poor system design 2% Problems with valve, positioners, actuators 3% Bad tuning 3% Process Performance is not as good as you think. D. Ender, Control Engineering More than 3% of installed controllers operate in manual More than 3% of the loops increase short term variability About 25% of the loops use default settings About 3% of the loops have equipment problems

9 PID versus More Advanced Controllers Error Past Present Future u(t) = k p ( βysp (t) y f (t) ) + k i t PI does not predict t t + T d ( ysp (τ) y f (τ) ) dτ + k d (γ dy sp dt Time dy ) f dt PID predicts by linear extrapolation, T d prediction horizon Advanced controllers predict using a mathematical model

10 Publications in Scopus 1 6 Pubications per year Year Number of publications by year for control (blue), PID (red) and model predictive control (green) from Scopus search for the words in title, abstract and keywords.

11 Outline 1. Introduction 2. Requirements 3. Tradeoffs 4. PI Control 5. PID Control 6. Relay Auto-tuners 7. Summary

12 Requirements d n y m y sp e u x F Σ C Σ P Σ y Controller 1 Process Attenuate load disturbances d Do not inject too much measurement noise n Robustness to model uncertainty Setpoint response - Can be dealt with separately by feedforward F (2 DOF, setpoint weighting, I-PD)

13 I PD Controller with filtering and antiwindup Filter y G f (s) ẏ f y f k d k p Σ u ff µ Actuator Model u r Σ e 1 s + k i Σ Σ e s k t The filter (can be combined with antialias filter) [ ] [ ][ ] d x1 1 x1 = + dt x 2 T 2 f T 1 f x 2 [ T 2 f ] y, has the states x 1 = y f and x 2 = dy f /dt. The filter thus gives filtered versions of the measured signal and its derivative. The second-order filter also provides good high-frequency roll-off.

14 Tune for Load Disturbances - Shinskey 1993 The user should not test the loop using setpoint changes if the set point is to remain constant most of the time. To tune for fast recovery from load changes, a load disturbance should be simulated by stepping the controller output in manual, and then transferring to auto. For lag-dominant processes, the two responses are markedly different. Process control: Tune k p, k i, k d and T f for load disturbances, measurement noise and robustness, then tune β, and γ for setpoint response. u(t) = k p ( βr(t) yf (t) ) + k i t Y f (s) = 1 1+sT f + s 2 T f 2 /2 Y(s) ( r(τ) yf (τ) ) dτ + k d (γ dr dt dy ) f dt

15 Assessment of Disturbance Reduction Compare open and closed loop systems! Y cl Y ol = 1 1+PC = S Geometric interpretation: Disturbances with frequencies outside are reduced. Disturbances with frequencies inside the circle are amplified by feedback, the maximum amplification is M s. Disturbances with frequencies less than sensitivity crossover frequency ω sc are reduced by feedback. 1 ω ms ω sc

16 Load Disturbance Attenuation Transfer function from load disturbance d to process outpur y ( P() = K ) G yd = P 1+PC = SP sp(s) s + Kki s s + Kki K s k i P = 2(s + 1) 4 PI: k p =.5, k i =.25 Gxd(ω) ω

17 Criteria and FOTD Model Traditionally the criteria IE = ITAE = e(t)dt, IAE = t e(t) dt, QE = e(t) dt, IE2 = (e 2 (t)+ ρu 2 (t))dt Notice that for a step u in the load disturbance we have The FOTD model u( ) = k i e(t)dt, IE = 1 k i P(s) = K 1+sT e sl, τ = L, τ 1 L+T e 2 (t)dt Lag dominant τ small (τ<.3) and delay dominant dynamics τ close to 1

18 Measurement Noise Injection d n C PID u Σ P x Σ y G g Controller Process Controller transfer function G f = 1 1+sT f + s 2 T 2 f /2 C PID(s) = k p + k i s +k ds, C = C PID G f Transfer function from measurement noise n to control signal u G un (s) = C 1+PC = SC s s + Kk i k i + k p s + k d s 2 s(1+st f +(st f ) 2 /2) Only controller parameters and K = P()

19 Bode Plots of Noise Transfer Function G un Lag dominated Balanced Delay dominated PI PID Validity of approximation (error in mid frequency range M s peak) Differences PI/PID lag dominated/delay dominated

20 Stochastic Modeling of Measurement Noise Measurement noise stationary with spectral density Φ(ω) σ 2 u = G un (iω) 2 Φ(ω)dω, σ 2 y f = G f (iω) 2 Φ(ω)dω k i + k p s + k d s 2 G un (s) (s + Kk i )(1+sT f +(st f ) 2 /2) ( ) σ u 2 k π i K + k2 p 2k i k d + 2 k2 d T f Tf 3 Φ, σ y 2 f = π Φ T f Noise gain k n = σ u /σ yf and SDU (standard deviation of u with white measurement noise Φ = 1) k i T f K + k2 p 2k i k d + 2 k2 d k nw = σ u σ yf ( πφ = 1 SDU = T 2 f k i K + k2 p 2k i k d T f + 2 k2 d T 3 f )

21 Outline 1. Introduction 2. Requirements 3. Tradeoffs 4. PI Control 5. PID Control 6. Relay Auto-tuners 7. Summary

22 Load Disturbance Attenuation and Robustness Performance (IAE= 1/k i blue) and robustness (M s, M t red) IE level curves are horizontal lines (P(s) = (s + 1) ki k p Little difference IE and IAE for k i <.4 and robust systems M s < 1.6 Approximately: k i gives performance and k p sets robustness

23 Load Disturbance Attenuation and Noise Injection 1 Process: P(s) = 1+.1s e s Controller: C = ( k p + k ) i s + k ds τ =.9 lag dominated! 1 1+sT f +(st f ) 2 /2 MIGO design without filtering: k p = 2.78, T i = 47.2, T d = 11.6 Filter time constants: T f = [ ] 1 2 IE k n

24 Outline 1. Introduction 2. Requirements 3. Tradeoffs 4. PI Control 5. PID Control 6. Relay Auto-tuners 7. Summary

25 Design Process Models - Essentially monotone step responses Ziegler-Nichols - Two parameters The FOTD model - Three parameters K,L, T G(s) = K 1+sT e sl L+T, τ 1 Lag (small τ ) and delay dominated dynamics τ close to one More complex models The test batch - essentially monotone dynamics Heritage of Eurotherm and Mike Sommerville 123 processes Normalized time delay τ = L y x Design controllers and match to model parameters

26 Constrained Optimization Modeling & control design Criteria Load disturbance attenuation IE, IAE Robustness M s M t Measurement noise SDU Noise gain k n Loop transfer function G l = PG f ( kp + k i s + k ds ) linear in parameters Many algorithms Convex-concave methods I L(iω) R L(iω) github.com/juliacontrol/controlsystems.jl/

27 PI Control: Minimize IAE, M s, M t Kk p vs τ 1 1 ak p vs τ T i /T vs τ T i /L vs τ The two parameter Ziegler-Nichols does not work (red dashed right figures)! Tuning of PI controller can be done with a three parameter FOTD model model

28 Some Tuning Rules Ziegler-Nichols step Ziegler-Nichols frequency k p =.9 K v L, k i =.27 K v L 3, T i = L/.3 k p =.45k u, k i =.54 k u T u, T i = T u /1.2 Lambda Tuning - T cl = T,2T,3T k p = T K(T cl + L), k 1 i = K(T cl + L), T i = T Skogestad SIMC Like Lambda but T i = min(t,4(t cl + L)) Skogestad SIMC+ AMIG (M s, M t = 1.4) k p = T + L/3 K(T cl + L), T i = min(t + L/3, 4(T cl + L)) k p =.15 K + (.35 LT (L+T) 2 ) T KL, T i =.35L+ 13LT 2 T LT + 7L 2

29 Tuning Lag-Dominated Dynamics 2 18 Lagdominant 1.9 S S ki λ A S S+ S S k p 1.6 Lambda tuning has very low gains S and S+ give similar tuning Lambda tuning gives constant integral time T i = k p /k i

30 Tuning Balanced Dynamics.4 3 Balanced ki S A S+ S S+ λ S S+ λ k p λ Tuning methods S+, A and λ gives similar results All controllers have constant integral time T i = k p /k i

31 Tuning Delay-Dominated Dynamics ki S S S 1.2 λ λ λ S+ S+ Delay dominated A S ZN k p Lambda tuning too high integral gain Obvious why Skogestad modified his method All controllers have constant integral time T i = k p /k i

32 Outline 1. Introduction 2. Requirements 3. Tradeoffs 4. PI Control 5. PID Control 6. Relay Auto-tuners 7. Summary

33 Difficulties with Derivative Action Shapes of stability region - don t fall off the cliff ki k p k d Filtering necessary

34 Temperature Control P(s) = e s C PI (s) = C PID (s) = s s s IE =.86, IAE =.1 IE =.21, IAE =.31 System output, y(t) Control signal, u(t) I L(iω) R L(iω)

35 IE or IAE for P = (s + 1) 3 C IE = s s IAE =.74 C κ = s s IAE =.57 C IAE = s 2.5 s 3 IAE =

36 PID Control: Minimize IAE, M s, M t Kk p vs τ 1 2 ak = k p KL/T vs τ T i /T vs τ T i /L vs τ T d /T vs τ T d /L vs τ Tuning rules based on FOTD can be found for τ>.3 More complex models for lag dominated dynamics Limiting cases K 1+sT e sl and K (1+sT/2) 2 e sl

37 Modeling for PI & PID Control AMIGO Tuning - complete testbatch 1 2 k i [PID]/k i [PI] vs τ circles: P(s) = K 1+sT e sl, squares: P(s) = K (1+sT) 2 e sl FOTD OK for τ>.4 better model required for smaller τ! Derivative action small improvement for τ>.8

38 Outline 1. Introduction 2. Requirements 3. Tradeoffs 4. PI Control 5. PID Control 6. Relay Auto-tuning 7. Summary

39 Relay Auto-tuning 1.5 y Relay feedback creats oscillation at ω 18! Automation of ZN frequency response method modified ZN tuning rules t

40 The First Industrial Test 1982

41 Temperature Control of Distillation Column

42 Commercial Autotuners One-button autotuning Three settings: fast, slow, delay dominated Automatic generation of gain schedules Adaptation of feedback gains Adaptation of feedforward gain Many versions Single loop controllers DCS systems Robust Excellent industrial experience Large numbers

43 Industrial Systems Functions Automatic tuning AT Automatic generation of gain scheduling GC Adaptive feedback AFB and adaptive feedforward AFF Sample of products NAF Controls SDM DCS AT, GS SattControl ECA SLC AT, GS Satt Control ECA SLC AT Alfa Laval Automation Alert DCS AT, GS Satt Control SattCon PLC AT, GS Satt Control ECA LC AT, GS, AFB, AFF Fisher Control DPR SLC Satt Control SattLine DCS AT, GS, AFB, AFF Emerson Delta V DCS AT, GS, AFB, AF ABB 8xA - 24 DCS AT, GS, AFB, AFF

44 Next Generation of Autotuners Observations A sine-wave input permits estimation of only two parameters PI controllers can be designed based on an FOTD model Little difference between PI and PID for processes with delay dominated dynamics Improvement by derivative action a factor 2 for τ =.45 PID controllers require better modeling if τ<.4 Separate real delays from higher order dynamics Suitable model classes Requirement on an auto-tuner Good excitation - modify relay and experiments Short experiment time - do not wait for steady state Other types of inputs - asymmetric relay additional inputs Trade-off buttons - performance & robustness related

45 Models Two parameter models Three parameter models P(s) = P(s) = P(s) = b, P(s) = K e sl s + a b s 2, P(s) = b + a 1 s + a 2 s + a e sl, P(s) = K K (1+sT) 2 e sl Four parameter models P(s) = b 1s + b 2 s 2 + a 1 s + a 2, P(s) = Five parameter model P(s) = b 1s + b 2 s 2 + a 1 s + a 2 e sl b s 2 + a 1 s + a 2 e sl 1+sT e sl

46 Typical Experiments u y u y u y Time [s] Figure from Josefin Berner

47 Better Excitation with Asymmetric Relay.8 U 2/ U ω [rad/s] Symmetric relay blue Asymmetric relay red Figure from Josefin Berner

48 Chirp Signal Broadband Excitation u(t) = (a+b t) sin(c + d t)t Frequency varies between a and c + d t max amplitude between a+b t max 4 u(t) t u(t)dt Notice both high and low frequency excitation t

49 Asymmetric Relay and Chirp Asymmetrical relay experiment combined chirp signal experiment Double experiment time. Constant amplitude, L =.1, w = 15 (1+.5 t), t max = 2.7,.15 ω L.35 Relay only P(s)=b exp( sl)/(s 2 +a1*s+a2) Relay and Chirp b exp( sl)/(s 2 +a1 s+a2)

50 Outline 1. Introduction 2. Requirements 3. Tradeoffs 4. PI Control 5. PID Control 6. Relay Auto-tuners 7. Summary

51 Summary Insight into PID control PI control can be designed based on FOTD model Importance of lag and delay dominant dynamics and normalized time delay τ PI is sufficient for delay dominated processes τ>.8 Derivative action helps for τ.8 Derivative action gives significant improvement for τ<.4 but improved models are required Next generation of relay auto-tuners Use system identification and model testing Use algorithms instead of simple tuning rules Admits tuning knob