EECE 460 : Control System Design


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1 EECE 460 : Control System Design PID Controller Design and Tuning Guy A. Dumont UBC EECE January 2012 Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
2 Contents 1 Introduction 2 Control Specifications 3 Empirical Tuning Methods ZieglerNichols CohenCoon Method The Good Gain Method 4 ModelBased Methods The Dahlin Controller ltuning Haalman s Method Internal Model Control SIMCPID Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
3 Introduction PID Controller Design and Tuning Vast literature on the topic. There is a plethora of techniques As Åström and Hägglund put it:... there are many different types of control problems and consequently many different design methods. To only use one method is as dangerous as to only believe in empirical tuning rules Empirical, historical methods ZieglerNichols oscillation or frequency response method The CohenCoon method The Good Gain method Modelbased Tuning LambdaTuning Haalman s method Skogestad s IMC rules Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
4 Control Specifications Control Specifications For response to step setpoint changes, the criteria below have been often used Z Z Z ITAE = t e(t) dt; ITE = te(t)dt; ITSE = te 2 (t)dt Rise time Settling time Decay ratio Overshoot ratio Steadystate error Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
5 Control Specifications Control Specifications For attenuation of load disturbances IAE = Z 0 e(t) dt; IE = Z It can be shown that for a unit step disturbance 0 IE = T i K c Z e(t)dt; ISE = e 2 (t)dt 0 Sensitivity to measurement noise Transmission of measurement noise to control signal is G nu = C/(1 + G 0 C) Normally at high frequencies G 0 C 0 and G nu C. For a PID controller, the high frequency gain is K c (1 + N) 10K c Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
6 Control Specifications Control Specifications Gain (or amplitude margin) A m = 1 G 0 C(iw u ) where w u is the ultimate frequency, i.e. argg 0 C(iw u )= Phase margin f m = p + argg 0 C(iw c ) where w c is the crossover frequency, i.e. G 0 C(iw c ) = 1 Robustness to model uncertainty 1 M s = max 1 + G 0 (iw)c(iw) = max S(iw) where S is the sensitivity function p Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
7 Control Specifications Control Specifications 1/M s is the stability margin Source: K.J. Astrom and T. Hagglund, Advanced PID Control. Published with permission of ISA. c All rights reserved. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
8 Control Specifications Control Specifications A number of optimizationbased techniques use M s to perform the tuning Source: K.J. Astrom and T. Hagglund, Advanced PID Control. Published with permission of ISA. c All rights reserved. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
9 Empirical Tuning Methods ZieglerNichols ZieglerNichols Frequency Method Only valid for open loop stable plants and it is carried out through the following steps 1 Set the true plant under proportional control, with a very small gain. 2 Increase the gain until the loop starts oscillating. Note that linear oscillation is required and that it should be detected at the controller output. 3 Record the controller critical gain K p = K c and the oscillation period of the controller output, P c. 4 Adjust the controller parameters according to the Table 6.1 on next slide. There is some controversy regarding the PID parameterization for which the ZN method was developed, but the version described here is, to the best knowledge of the authors, applicable to the parameterization of standard form PID: C(s)=K p T r s + T ds 1 + t d s Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
10 Empirical Tuning Methods ZieglerNichols ZieglerNichols Frequency Method Source: K.J. Astrom and T. Hagglund, Advanced PID Control. Published with permission of ISA. c All rights reserved. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
11 Empirical Tuning Methods ZieglerNichols ZieglerNichols Frequency Method Consider a plant of the form G 0 (s)= K 0e st 0 n 0 s + 1 where n 0 > 0 Figure 6.3 from the textbook Response is very sensitive to the ratio between the delay and the time constant Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
12 Empirical Tuning Methods ZieglerNichols Interpretation of the Frequency Response Method The critical gain K c is such that the Nyquist plot of the loop gain goes through the 1 point at the oscillation frequency w c, i.e. K c G 0 (jw c )= 1 Method where one point of the Nyquist curve is moved Source: K.J. Astrom and T. Hagglund, Advanced PID Control. Published with permission of ISA. c All rights reserved. With PI control, ultimate point moved to (0.4,+0.08i) With PID control, ultimate point moved to (0.6,0.28i) Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
13 Empirical Tuning Methods ZieglerNichols Interpretation of the Frequency Response Method Generally better for PID than for PI Quarter amplitude decay ratio gives poorly damped closedloop system No tuning parameter Based on only one point on the Nyquist curve Major shortcoming of the ZN method is that it requires that the plant be forced to oscillate with a non predictable amplitude. This can be dangerous and expensive! Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
14 Empirical Tuning Methods ZieglerNichols Performance of ZieglerNichols Method Define the normalized gain of a process as: and the normalized dead time as: k = G 0(iw u ) G 0 (0) t = D D + T where D is the dead time and T is the dominant time constant Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
15 Empirical Tuning Methods ZieglerNichols Performance of ZieglerNichols Method Small k and t: Easy to control. Other methods usually give better performance than ZieglerNichols Intermediate k and t: Primary range of use for ZieglerNichols tuning. Setpoint weighting can reduce overshoot k and t close to 1: Dynamics dominated by dead time. ZieglerNichols should not be used. Actually, PID should probably not be used at all. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
16 CohenCoon Method Empirical Tuning Methods CohenCoon Method A simple model G 0 (s)= K 0e st 0 n 0 s+1 is built using the procedure below 1 With the plant in open loop, take the plant manually to a normal operating point. The plant at y(t)=y 0 for a constant u(t)=u 0. 2 At an initial time, t 0, apply a step change to the plant input, from u 0 to u (this should be in the range of 10 to 20% of full scale). 3 Record the plant output until it settles to the new operating point. Assume you obtain the curve shown below (m.s.t. stands for maximum slope tangent. This curve is known as the process reaction curve. 4 Compute the parameter model as follows: Figure 6.6 from textbook K 0 = y y 0 u u 0 ; t 0 = t 1 t 0 ; n 0 = t 2 t 1 Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
17 CohenCoon Method Empirical Tuning Methods CohenCoon Method Consider again a plant of the form G 0 (s)= K 0e st 0 n 0 s + 1 where n 0 > 0 CohenCoon propose the following tuning Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
18 CohenCoon Method Empirical Tuning Methods CohenCoon Method Figure 6.8 from the textbook Response is still quite sensitive to the ratio between the delay and the time constant Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
19 Empirical Tuning Methods The Good Gain Method The Good Gain Method inspired by ZieglerNichols oscillation method, but does not require the control loop to go into oscillations. 1 With the process close to the specified operating point, make sure the controller is a pure proportional controller. 2 Increase the proportional gain K p until the step response displays an overshoot with minimal undershoot. Note the value of the proportional as K PGG 3 Denote T ou as the time between the overshoot and the undershoot. For more details, see Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
20 Empirical Tuning Methods The Good Gain Method The Good Gain Method Good gain rules for PI K C = 0.8K PGG T I = 1.5T ou Good gain rules for PID K C = 0.8K PGG T I = 1.5T ou T D = T I 4 Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
21 Empirical Tuning Methods The Good Gain Method The Good Gain Method  Theoretical Background Assume that under K PGG the closedloop system behaves as a secondorder transfer function: Kw 2 0 s s + 2zw 0 s + w 2 0 z = 0.6 would give a 10% overshoot with minimal undershoot. The period of the damped oscillation is P GG = 2p p 1 z 2 w 0 = 2p q w0 2 = 2p 0.8w 0 = 2T ou In ZieglerNichols, P ZN = 2p/w 0, i.e. P ZN = 0.8P GG = 1.6T ou. Then, T I = P ZN /1.2 = 1.33T ou In GG, to increase robustness, T I = 1.5T ou. To compensate for integral action, K c is reduced to K c = 0.8K PGG Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
22 Dahlin Controller ModelBased Methods The Dahlin Controller Developed in 1968 by Dahlin and Higham Used extensively in industry in its digital form, particularly in paper machine control systems Consider a feedback control loop with a process P(s) and controller C(s). the closedloop system can be written as: Y(s)= P(s)C(s) 1 + P(s)C(s) Y 1 sp(s)+ 1 + P(s)C(s) W(s) with setpoint Y sp (s) and disturbance W(s). If D(s) denotes the desired closedloop transfer function then we want to solve for C(s) such that D(s)= P(s)C(s) 1 + P(s)C(s) Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
23 Dahlin Controller ModelBased Methods The Dahlin Controller The process transfer function P can be factored into two parts: P N which contains the dead time and the righthalf plane or poorly damped zeros, i.e. elements which cannot be cancelled by the controller P M which contains the minimum phase elements that can be cancelled by the controller Thus D(s)=P N (s)d (s) where D (s) is the arbitrary portion of the desired closedloop transfer function and is usually chosen as D (s)= ls Solving for the controller C(s) then gives Dahlin Controller C(s)= 1 D (s) P M (s) 1 P N (s)d (s) Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
24 Predictive PI Control ModelBased Methods The Dahlin Controller Consider the process P(s)= K p 1 + Ts e sl The desired closedloop transfer function is The Dahlin controller is then Dahlin controller for FOPDT C(s)= D(s)= e sl 1 + ls 1 + st K p (1 + ls e sl ) This can be interpreted as a PI controller with deadtime compensation. When L = 0, this becomes a simple PI controller. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
25 ModelBased Methods Lambda Tuning for FOPDT ltuning Recently popularized in pulp and paper industry by Bialkowski Based on the Dahlin algorithm Dead time approximated by rational transfer function If dead time L is approximated by e sl 1 sl the controller is which is a PI controller with PI ltuning for FOPDT C(s)= 1 + st K p (l + L)s T i = T and K c = T K p (l + L) Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
26 ModelBased Methods Lambda Tuning for FOPDT ltuning If dead time L is approximated by the controller is C(s)= e sl 1 sl/2 1 + sl/2 (1 + sl/2)(1 + st) K p s(l + l + sll/2) which can be approximated by a PID controller PID ltuning for FOPDT C PID (s)= (1 + sl/2)(1 + st) K p s(l + l) Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
27 ModelBased Methods ltuning Typical Recommendations for Process Industries Bialkowski 1 makes the following recommendations when using ltuning: Flow control: l = two to three times the process time constant Temperature control: PID tuned with l slightly smaller than larger process time constant Consistency control: l > process time constant plus dead time Tank level control: l three dead times This is actually a special case of pole placement, to be seen in more details very soon! A drawback of ltuning is that it cancels the process poles, which can give poor load disturbance rejection characteristics 1 N.J. Sell (Ed.), Process Control Fundamentals for the Pulp & Paper Industry, Tappi Press Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
28 Haalman s Method ModelBased Methods Haalman s Method Haalman has suggested choosing an "ideal" loop transfer function G l = PC and then computing the controller C = G l /P. Haalman suggests choosing This gives M s = 1.9 G l (s)= 2 3Ls e sl Note that only the dead time influences the loop transfer function. All process poles and zeros are cancelled which might lead to difficulties Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
29 Haalman s Method ModelBased Methods Haalman s Method Given the process Haalman s method gives P(s)= K p 1 + st e sl C(s)= 2(1 + st) 3K p Ls = 2T 3K p L (1 + 1 st ) which is a PI controller with K = 2T/3K p L and T i = T. (ZN would give K = 0.9T/L and T i = 3L) Haalman s method and ltuning would give the same when T cl = L/2 Haalman s method more reasonable when time delay L is large Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
30 Haalman s Method ModelBased Methods Haalman s Method Source: K.J. Astrom and T. Hagglund, Advanced PID Control. Published with permission of ISA. c All rights reserved. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
31 Haalman s Method ModelBased Methods Haalman s Method A problem with Haalman method is that it cancels all the poles and zeros. Cancelling all poles and zeros may be bad. Consider the plant G(s)=e sl /(1 + st) with the PI controller C(s)=K(1 + st)/st, then we can write du(t) dt dy(t) dt dy(t) = K dt = 1 (u(t L) y(t)) T + y(t) = K u(t L) T T With initial conditions y(0)=1 and u(t)=0 for L < t < 0 the openloop response is y o (t)=e t/t In closedloop, equations above show that u(t)=0, hence y cl = y ol The controller does nothing to reduce the error! We will discuss this phenomenon in more details when presenting the Qdesign This is a problem for all design methods that cancel all process poles. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
32 Haalman Method ModelBased Methods Haalman s Method Source: K.J. Astrom and T. Hagglund, Advanced PID Control. Published with permission of ISA. c All rights reserved. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
33 ModelBased Methods Internal Model Control Procedure Internal Model Control The IMC is a general control design procedure It factorizes the plant model G 0 (s) into an invertible minimumphase G m (s) part and a noninvertible allpass G a (s) part We then choose G 0 (s)=g m (s)g a (s) T(s)=F(s)G a (s) where F(s) is a lowpass filter typically of the form F(s)=1/(t c s + 1) n Knowing that T = G 0 C/(1 + G 0 C) we can solve for C IMC Controller C(s)=G m (s) 1 1 F(s) G a (s) By making assumptions about G 0 (s), we can obtain PI and PID controllers. Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
34 Skogestad s IMC Rules ModelBased Methods SIMCPID SIMC Rules According to the author, "probably the best simple PID tuning rules in the world" a a S. Skogestad. Simple analytic rules for model reduction and PID controller tuning. J. Process Control, 13 (2003): Given a closedloop system, the complementary sensitivity function T(s) is specified as T(s)= C(s)G(s) 1 + C(s)G(s) = t c s e ts With G(s)=K/(1 + t 1 s)(1 + t 2 s), where t 1 >> t 2 and approximating the delay as e ts = 1 ts, this gives a seriesform PID: C(s)= (1 + t 1s)(1 + t 2 s) K(t c + t)s Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
35 Skogestad s IMC Rules ModelBased Methods SIMCPID The straight IMC rule thus gives K c = 1 K t 1 (t c + t) = 1 1 K 0 (t c + t) ; T I = t 1 ; T D = t 2 A problem with this choice of T I is that although it works well for set point changes, when t 1 >> t, it is sluggish in rejecting load disturbances. Skogestad thus proposes to instead choose T I as T I = min(t 1,4(t c + t)) Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
36 Skogestad s IMC Rules ModelBased Methods SIMCPID Table: SIMC PID Tuning Rules 2 G(s) K c T I T D Ke ts 1 t 1 K (t c +t) min(t 1,4(t c + t)) (1+t 1 s) Ke ts (1+t 1 s)(1+t 2 s) Ke ts s Ke ts s(1+t 2 s) 1 K 1 K 1 K Ke ts s 2 1 K t 1 (t c +t) min(t 1,4(t c + t)) t 2 1 (t c +t) 4(t c + t) 1 (t c +t) 4(t c + t) t 2 1 4(t c +t) 2 4(t c + t) 4(t c + t) 2 The derivative time is for the series form PID controller Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
37 Summary ModelBased Methods SIMCPID Limit PID to control of loworder systems and systems with small delay Modelbased tuning methods are preferrable to empirical ones Dahlin control design, Haalman s method and IMC are simple to use but cancel all process poles These are special cases of more general design techniques Pole placement Qdesign We shall study those techniques in details Guy A. Dumont (UBC EECE) EECE 460 PID Tuning January / 37
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