Unified PID Tuning Approach for Stable, Integrative, and Unstable

Article pubs.acs.org/IECR

Unified PID Tuning Approach for Stable, Integrative, and Unstable Dead-Time Processes⊥ Julio E. Normey-Rico† and José Luis Guzmán*,‡ †

Departamento de Automaçaõ e Sistemas, Federal University of Santa Catarina, Florianópolis, Santa Catarina, Brazil Departamento de Informática, Universidad de Almería, Almeria, Spain

‡

ABSTRACT: This paper presents a unified approach, which is based on a PID approximation of the filtered Smith predictor, for tuning PID controllers for stable, integrative, and unstable dead-time processes. The proposed control tuning method is simple to analyze and use. Case studies are included to illustrate the advantages of the proposed tuning rules. Comparisons with other existing methods are also presented to show that the proposed unified method provides promising results. Furthermore, tuning rules to obtain a reasonable trade-off between robustness and performance are derived.

■

INTRODUCTION

integrative and unstable processesusing a unique design approach. Thus, this paper presents a PID tuning procedure that can be used to control FOPDT, IPDT, and UFOPDT processes. The proposed PID is based on a simple modification of the ideal dead-time compensation structure of the filtered Smith predictor (FSP),19−21 in which both the design and tuning of the controller are intuitive and simple. In fact, the proposed method has been developed with the objective of looking foremost at the objectives proposed by Skogestaad in ref 3: (i) the tuning rules are well motivated, model-based, and analytically derived; (ii) they are simple; and (iii) they work well on a wide range of processes. Furthermore, the proposed method considers a real PID controller, which includes the filter of the derivative action in the design. The resulting PID controllers have only one tuning parameter, which is very intuitive and easy to tune, trying to find a trade-off between robustness and performance. In this way, first, the proposed PID controllers were compared with well-known PID tuning rules through a simulation study, in which case the tuning parameter was set manually to obtain responses similar to those obtained with the other tuning methods. Afterward, a tuning methodology is presented to derive simple guidelines for the tuning parameter considering both robustness and performance specifications. For FOPDT and IPDT processes, robustness is measured using the sensitivity peak, and a robustness index based on the delay margin is used in the unstable case. The output performance is quantified by using the integral absolute error (IAE) measurement for both set point tracking and load disturbance rejection problems, in which the IAE value obtainend with the FSP (previously designed for the same robustness properties) is used as the performance target. The paper is organized as follows. The section below is devoted to describing the controller design and the resulting

PID controllers are widely used in industry, mainly in process control applications, where more than 95% of the controllers are of the PID type,1 such as was recently corroborated in the last IFAC PID Conference.2 As pointed out in refs 3 and 4, although a PID controller has only three parameters, it is not easy, without a systematic procedure, to find adequate values (settings) for them. In industry, a large number of PID controllers are poorly tuned.1 Despite the development of other more powerful control algorithms and the fact that hundreds of tuning rules for PID controllers have been proposed since 1942, every year, new works are developed aiming for the improvement of PID tuning.5 It is important to note that most of the proposed tuning rules are based on simple models, such as a stable first-order plus dead time model (FOPDT), an integrator plus dead time model (IPDT), or an unstable first-order plus dead time model (UFOPDT). This is motivated by the fact that these simple models are easy to obtain in industry from process data. Many of these tuning rules are derived for only FOPDT models (see, for example, refs 6−8), for only IPDT models (see, for instance, refs 9, 10), or for only UFOPDT models (see, for example, refs 11−14). In some cases, the proposed rules are valid for two cases: FOPDT−IPDT,3,4,15,16 IPDT−UFOPDT,17 or FOPDT−UFOPDT.18 Moreover, different control objectives are used to obtain the PID parameters. For example, in refs 15 and 6, settings result in a very good disturbance response for integrating processes but give aggressive settings for set-point changes and poor performance for dominant deadtime processes. Robustness and good set-point responses are obtained with the IMC approach,7 however, with slow disturbance rejection in lag dominant plants. More recently, a simple tuning was proposed3,4 that gives satisfactory set-point and disturbance responses for stable and integrative cases. On the other hand, the AMIGO method proposed in ref 16 provides simple tuning rules for stable and integrative deadtime processes looking for a compromise between robustness and performance. However, there are not too many time domain tuning rules to be applied for all casesstable, © 2013 American Chemical Society

Received: Revised: Accepted: Published: 16811

May 30, October October October

2013 25, 2013 29, 2013 29, 2013

dx.doi.org/10.1021/ie401722y | Ind. Eng. Chem. Res. 2013, 52, 16811−16819

Industrial & Engineering Chemistry Research

Article

Figure 1. FSP: structure for analysis.

• UFOPDT: τi = T0(2 + T0/T), K = (T0 + 2T)/(T0Kp) Filter Fr(s) is designed to avoid the effect of the open-loop pole in the disturbance closed-loop response. This allows for an internally stable system that rejects steps in steady state with time constant T0.20 The obtained filter is • FOPDT and IPDT: Fr(s) = ((1 + sβ)/(1 + sT0)) • UFOPDT: Fr(s) = ((1 + sβ)/(1 + sT0(2 + T0/T))) where β is obtained for the specified conditions and T0 is the only tuning parameter for a trade-off between robustness and performance. Note that increasing T0 a slower and more robust system is obtained.20 Finally, a reference filter, F(s) = (1 + sTr)/ (1 + sτi), can be used, if necessary, to eliminate the effect of the controller zero. In this filter, Tr regulates the overshoot of the set-point response. With these settings, the FSP allows one to speed up the disturbance response of the SP, even for the case of lagdominant processes and to control integrative and unstable processes. In practice, the FSP equivalent 2DOF controller shown in Figure 2 is used for implementation purposes.

tuning rules for FOPDT, IPDT, and UFOPDT systems. First, the FSP scheme is briefly described, and afterward, the equivalent PID controller is obtained, which is then used to derive the resulting tuning rules. Three simulated case studies are presented in the next section, where the proposed tuning rule is compared with other existing methods. Then, the following section describes the tuning procedure used to obtain guidelines to find a reasonable trade-off between robustness and performance. Finally, the paper ends with some conclusions.

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THE PROPOSED METHOD

A unified dead-time compensation structure that can be used to control stable, integrative, and unstable processes with a dead time has been recently proposed.20 The structure is based on a simple modification of the Smith predictor (SP), and both the design and tuning of the controller are simple and allow faster closed-loop responses than other controllers. In practice, the use of a dead-time compensator is the best solution for a deadtime process; however, sometimes the only real-time allowable controller is a PID, and a simple first-order model (FOPDT, IPDT, or UFOPDT) is used to represent the process. Therefore, the idea of this work is based on two steps: first, a FSP is computed for a first-order model, and then the equivalent FSP control law is approximated by a PID. Furthermore, the resulting tuning rule is derived to eliminate the slow or unstable dynamics from the disturbance rejection responses. FSP for FOPDT, IPDT, and UFOPDT Processes. This section briefly revises the FSP tuning for FOPDT, IPDT, and UFOPDT processes. The FSP controller is shown in Figure 1.20 As can be seen, the structure is the same as in the SP with two additional filters. F(s) is a traditional reference filter to improve the set-point response and Fr(s) is a predictor filter used to improve the predictor properties. In the structure used for analysis, Pn(s) = Gn(s)e−Lns is a model of the process, Gn(s) is the dead-time-free model, and C(s) a PI primary controller, C(s) = K((1 + sτi)/sτi), where K and τi are the proportional gain and the integral time, respectively. For the stable case, Gn(s) = (Kp/(1 + sT)), for the integrative case, Gn(s) = Kp/s, and for the unstable case, Gn(s) = (Kp/(sT − 1)). In the three cases, the PI controller is tuned to obtain a delay free nominal closed-loop system with a pole (or double pole) in s = −1/T0, giving

Figure 2. 2DOF equivalent structure of the FSP.

This 2DOF controller is given by Ceq(s) =

C(s) Fr(s) 1 + C(s) Gn(s) (1 − e−Lns Fr(s))

Feq(s) =

,

F (s ) Fr(s)

and it is usually implemented in the discrete domain20 in such a way that Ceq(s) does not cancel the open-loop pole of the model. PID Approximation of the FSP. As mentioned, if only a PID controller is allowable, the exact FSP solution cannot be used. Thus, the idea of this paper is to approximate, at low frequencies, the FSP by a PID. To do it, first, the equivalent Ceq(s) controller of the FSP is computed for the three simple cases: FOPDT:

• FOPDT: τi = T, K = T/(T0Kp) • IPDT: K = 1/T0Kp (P controller)

Ceq(s) = 16812

K (1 + sβ)(1 + sT ) sτi(1 + sT ) + KK p[1 + sτi − e−Lns(1 + sβ)] dx.doi.org/10.1021/ie401722y | Ind. Eng. Chem. Res. 2013, 52, 16811−16819


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Figure 3. Examples of PID approximation of the FSP for stable, P(s) = e−0.5s/(s + 1); integrative, P(s) = e−s/s; and unstable, P(s) = e−0.4s/(s − 1), cases. Dashed lines represent the FSP frequency responses, and the solid lines, the PID approximations.

IPDT:

T0

α=

Ceq(s) =

Ln + T0 +

K (1 + sβ)s s(1 + sT0) + KK p[1 + sT0 − e−Lns(1 + sβ)]

LnTi 2T0

−

Ti = 2T0 + Ln ,

K (1 + sβ)(1 − sT ) sτi(1 − sT ) + KK p[1 + sτi − e−Lns(1 + sβ)]

kc(1 + sTi)(1 + sTd) sTi(1 + sαTd)

kc =

Ti K p(Ln + 2T0 − Ti)

2 + Ln /T0 , K pδ

α=

T0 δ

(5) (6)

UFOPDT: Ti =

T0(2T + T0)(2 + Ln /T ) + 2LnT 2T − Ln

kc =

TT i , K pδ

α=

T0 2 δ

δ = T0 2 − T (L + T0(2 + T0/T ) − Ti)

(7)

(8) (9)

In the unstable case, the solution is valid only for processes with L < 2T. Notice that this is not a very hard constraint because in practice, it is difficult to find dominant dead-time unstable processes (this point is discussed with more details later on in PID Approximation of the FSP). Moreover, as shown in ref 19, dominant dead-time unstable processes can be unstabilized with an infinitesimal dead-time error, even using an ideal dead-time compensator structure. To show that the proposed PID approximation of the FSP performs properly, Figure 3 shows an example of the PID approximations for the stable, integrative, and unstable cases (these examples correspond to the three case studies presented in FSP for FOPDT, IPDT, and UFOPDT Processes). As can be seen in the Bode diagrams, the resulting PID controllers

(1)

where Td = 0.5Ln, and Kc, Ti, and α are given by FOPDT: ⎡ (2T − Ln)(T − T0)2 ⎤ ⎥ Ti = T ⎢1 − (2T + Ln)T 2 ⎦ ⎣

kc =

δ = T0 + 0.5Ln(4 + Ln /T0)

Note that with the proposed tuning rule, the term KKp[1 + sT0 − e−Lns(1 + sβ)] in the denominator of Ceq(s) has a root at s = 0, and, in the IPDT case, its derivative gain is −1 in such a way that the denominator of Ceq(s) has two roots at s = 0. Thus, the minimal expression of Ceq(s) has integral action in all cases. To obtain a PID controller, Cpid(s), that approximates Ceq(s), a Pade approximation of the dead time is used, e−Lns = ((1−0.5Lns)/(1 + 0.5Lns)). Afterward, parameter β is calculated to eliminate the zero of Ceq(s) at the desired value; that is, s = −1/T in the stable case, s = 0 in the integrative case, and s = 1/ T in the unstable case. The final real PID series controller (a realizable transfer function) is Cpid(s) =

(4)

IPDT:

UFOPDT: Ceq(s) =

T0Ln 2T

(2)

(3) 16813



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Figure 4. Stable case. Example for transfer function P(s) = e−0.5s/(s + 1) for a reference value of r(t) = 1, ∀t ≥ 0 and a load disturbance value of q(t) = 0.3, ∀t ≥ 12.5. A dead-time modeling error of 10% was used for robustness analysis.

approximates very well the Ceq of the FSP in low frequencies, such as expected.

time delay processes and for both set points and load disturbances. Stable Case. For the FOPDT case, an example from ref 18 is selected in which the process transfer function is given by

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CASE STUDIES This section is devoted to analyzing the three case studies, FOPDT, IPDT, and UFOPDT, where the results obtained for the proposed tuning rule are compared with other algorithms found in literature. For all the examples, the time unit is considered in seconds. As discussed at the Introduction, there exist many methods providing PID tuning rules for FOPDT, IPDT, and UFOPDT systems (although none of them give a general PID tuning for the three cases). Here, the time-domain methods proposed for FOPDT and UFOPDT,18 for FOPDT,17 for IPDT and UFOPDT,22 and for FOPDT and IPDT3,4 have been selected considering the same features of the rule proposed in this paper: simple tuning rule based on the process parameters and easy understanding design procedure. Furthermore, these four works include comparisons with other existing methods, and thus, they are quite valuable to be used as reference tests and to obtain adequate conclusions. The method proposed in ref 18 presents a tuning rule that is based on designing a standard PID controller that looks to match the numerator and denominator of the closed-loop transfer function. Simple equations (with no tuning parameter) are derived for stable and unstable processes, but the proposed rule for unstable processes is limited to L/τ ≤ 1.2. The IMC-PID approach presented in ref 7 is generalized in ref 17 to show how to obtain PID parameters for general stable process models. The standard PID controller is obtained by taking the first three terms of the Maclaurin series expansion of the single-loop form of the IMC controller. The work presented in ref 22 proposes a method that is an extension of the PID controller tuning presented in 17 to general unstable and integrating processes with time delay. Explicit PID controller tuning rules based on IMC are proposed for unstable and integrative processes with time delay. On the other hand, Skogestad proposed a method based on analytic rules for series PID controller tuning that are simple and still result in good closed-loop behavior.3,4 They are based on the IMC-PID tuning rules, in which the rule for the integral term is modified to improve disturbance rejection for integrating processes. The resulting tuning rule works well for both integrating and stable

P(s) =

e−0.5s s+1

(10)

The methods described above for stable processes were used for looking for obtaining a similar rise time (0.8 s) for the set point tracking response. This decision was made because the method proposed by ref 18 does not provide any tuning parameter, and thus, the other methods were tuned accordingly to have a similar rise time. In this way, the method proposed in ref 18 gives a controller with kc = 2.5, Ti = 1.25, and Td = 0.217; that proposed in ref 17 results in kc = 2.01, Ti = 1.21, and Td = 0.18 with λL = 0.1 (where λL is the closed-loop tuning parameter); the tuning rule from ref 4 provides kc = 1.56, Ti = 1.16, and Td = 0, with τc = 0.25 (with τc being the closed-loop tuning parameter); and the parameters for the proposed method according to eqs 2−4 are kc = 1.94, Ti = 0.784, Td = 0.25, and α = 0.31 with T0 = 0.4. For this comparison, a reference filter was not used. Figure 4 and Table 1 show the graphical and numerical results for the different tuning rules, where in the table OS is Table 1. Numerical Results for Stable Case structure

OS (%)

IAEr

ITAEr

tr

IAEd

ITAEd

Ms

proposed Padma et al.18 Lee et al.17 Skogestad3

25.32 16.98 30.20 48.02

3.93 3.97 4.08 4.46

7.68 7.87 8.58 10.64

0.80 0.81 0.80 0.81

0.52 0.60 0.52 0.63

0.87 1.20 1.10 1.31

2.18 3.02 2.04 2.09

the overshoot of the set-point response, IAEr is the integrated absolute error for the reference tracking, IAEr is the integrated time weighted absolute error for the reference tracking, tr is the rise time for the reference tracking, and IAEd and ITAEd are equivalent measurements to IAEr and ITAEr for the load disturbance response. For the robustness analysism two indexes are used. The sensitivity peak, Ms, which is the peak of the sensitivity function in the frequency domain, defined as 16814



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Figure 5. Integrative case. Example for transfer function P(s) = e−s/s for a reference value of r(t) = 1, ∀t ≥ 0 and a load disturbance value of q(t) = 0.3, ∀t ≥ 30. A dead-time modeling error of 10% was used for the robustness analysis.

Table 2. Numerical Results for Integrative Case structure

OS (%)

IAEr

ITAEr

tr

IAEd

ITAEd

Ms

proposed Lee et al.22 Skogestad3

1.04 2.10 0.59

5.81 6.00 5.81

18.60 20.40 18.90

7.80 7.90 12.7

13.2 13.67 14.46

89.90 96.35 110.42

1.59 1.50 1.76

Ms = max ω

1 |Cpid(jω) Pn(jω)|

load disturbance rejection based on the IAE and ITAE values presented in Table 1. Finally, notice that from the robustness analysis shown in Figure 4, where a dead-time error of 10% was considered, the unified proposed method is that with better robustness properties. However, considering Ms as a measurement of robustness, the proposed method is equivalent to the methods in 17 and 3. Integrative Case. The following transfer function has been used for the IPDT case:3

∀ω>0

where Pn(s) is the nominal model of the plant used to tune the controller. In the graphical analysis, a robustness index, RI(ω), defined as RI(ω): =

|1 + C(jω) Pn(jω)| |C(jω) Pn(jω)|

∀ω>0

(11)

and is used to verify the robust stability condition RI(ω) > dP (jω),∀ω > 0, where dP (ω) is the multiplicative norm-bound uncertainty, the real plant is P(ω) = Pn(ω)[1 + dP(ω)], dP (ω) ≥ dP(ω), and it is considered that the nominal system is stable.23 Notice how all the methods reach essentially the same rise time and similar load disturbance responses are obtained for the three methods. The main differences are observed at the set point tracking response, where an oscillatory behavior is obtained for all the methods, as expected. The minimum and maximum overshoots are reached for the methods in refs 17 and 3, respectively. On the other hand, the method proposed in this paper obtains better responses for set point tracking and

P(s) =

e −s s

(12)

In this case, the resulting PID parameters are kc = 0.502, Ti = 6.87, and Td = 0.219 with λL = 2.75 for the method in ref 22; kc = 0.526, Ti = 7.6, and Td = 0 with λS = 0.9 for the method in ref 3; and kc = 0.48, Ti = 6.4, Td = 0.5, and α = 0.56 with T0 = 2.75, for the proposed tuning rule following eqs 5 and 6. The different free tuning parameters were modified to obtain similar responses for set point tracking and load disturbance responses. In this example, a fist-order reference filter has been used for the three methods to obtain a underdamped response with F(s) = (3.84s + 1)/(6.4s + 1). 16815



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Figure 6. Unstable case. Example for transfer function P(s) = e−0.4s/(s − 1) for a reference value of r(t) = 1, ∀t ≥ 0 and a load disturbance value of q(t) = 0.3, ∀t ≥ 12.5. A dead-time modeling error of 10% was used for the robustness analysis.

The graphical responses and numerical results are presented in Figure 5 and Table 2, respectively. Notice how nearly the same response is obtained for the three methods. Slight differences can be observed in Table 2, where it can be seen that the method in ref 3 is the slowest one for the load disturbance response, although it is also the method with less overshoot for the set point tracking response. On the other hand, the proposed method again gives the best results according to the IAE and ITAE measurements, although as pointed out before, there are only some minor differences among the three methods. Regarding the robustness of the analysis presented in Figure 5, the three methods also showed similar robustness properties, yielding for both the proposed method and the Skogestad approach essentially the same results. However, considering Ms, the proposed method provides a better robustness result than the Skogestad one. Unstable Case. Finally, a third example for the UFOPDT system in which the process transfer function is described by ref 22 is presented. P(s) =

e−0.4s s−1

Table 3. Numerical Results for Unstable Case structure

OS (%)

IAEr

ITAEr

tr

IAEd

ITAEd

Ms

proposed Padma et al.18 Lee et al.22

9.90 13.00 9.07

3.88 4.02 3.91

7.53 8.34 7.63

1.4 2.10 1.5

0.71 0.86 0.75

1.14 1.58 1.26

4.37 3.57 3.42

tuning parameter that allows one to define the desired closedloop response. This fact is not available using the tuning rule of ref 18, as mentioned above. On the other hand, the three methods have similar robustness capabilities, as shown in Figure 6 by means of RI, the proposed unified method being slightly more robust, however, because Ms is the one that presents the best value in ref 22. The analysis of this section had as an objective to show how the proposed unified PID tuning approach can give a similar or better performance than others PID tuning rules for some particular cases. However, in practice, this comparative analysis is not useful, and some guidelines need to be developed to choose the free parameter, T0, of the PID. Thus, for completeness, the next section discusses how to tune T0 for a given set of specifications.

(13)

The method proposed in ref 18 consists of a direct design tuning rule without using any design parameter. Thus, and for comparative reasons and to reach an adequate conclusion, the other methods have been tuned trying to obtain similar load disturbance responses. The resulting controller parameters are kc = 2.75, Ti = 2.22, and Td = 0.21 for the method in ref 18; kc = 2.90, Ti = 2.10, and Td = 0.16 for the method in ref 22 with λL = 0.4; and kc = 1.07, Ti = 1.86, Td = 0.2, and α = 0.21 with T0 = 0.4 for the proposed tuning rule by using eqs 7−9. Again, a first-order reference filter was used for the three methods to reduce the overshoot in the set point tracking response with F(s) = (0.46s + 1)/(1.86s + 1). Figure 6 and Table 3 present the results for this case. It can be observed how the responses for the method in ref 22 and the proposed tuning rule yield quite similar results. This fact can be better seen from the performance measurements in Table 3, where the values for both methods are essentially identical. Furthermore, these two methods not only improve the results with respect to the method of ref 18 but also provide a design

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PID TUNING METHODOLOGY This section presents the methodology used for the tuning of parameter T0 in the proposed PID controller. The tuning strategy considers both robustness and performance specifications. To quantify the output performance, the IAE measurement is used. Robustness is measured in two different ways: for the stable and integrative cases, considering the sensitivity peak, and for unstable plants, using a robustness index based on the delay margin. Tuning for Stable and Integrative Plants. As pointed out above, for these cases, the sensitivity peak, Ms is used to measure the closed-loop robustness. The output performance is quantified using the IAE value: IAE = 16816

∫0

∞

|y(t ) − r(t )| dt dx.doi.org/10.1021/ie401722y | Ind. Eng. Chem. Res. 2013, 52, 16811−16819


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computed for both reference changes, IAEr, and disturbance rejection, IAEd, respectively. Because the proposed PID controller is derived from the FSP controller by means of a polynomial approximation of the delay, the IAE values obtained with FSP are used as the ideal achievable performance. Thus, the objective of the proposed tuning rule is to find the best value of T0 that gives a trade-off between robustness (for a desired Ms) and performance, for which the ideal performance is that given by the FSP tuned for the same Ms value. Therefore, the following steps are executed in the design procedure: • Choose a desired Ms = M0s • Tune T0 in the FSP to achieve the desired M0s • Compute the IAEr and IAEd for the FSP controller and define these values as optimal indexes, IAE0r and IAE0d • First tuning: only robustness is considered. T0 is tuned in the PID to minimize the following criterion: min J(T0) = min{[Ms0 − Ms(T0)]2 } T0

T0

Figure 7. T0 tuning rule for the stable case with Ms = 1.6 for the robustness criterion (circles) and for the performance−robustness trade-off (solid dots).

• Second tuning: Trade-off between robustness and performance. Tune T0 in the PID to minimize the following criterion:

FSP for FOPDT, IPDT, and UFOPDT Processes, where L/T = 0.5, the two solutions shown in Figure 7 give approximately the same result, T0 = 0.55 for Ms = 1.6. As in that example, T0 = 0.4 was selected, and the obtained PID controller has an Ms > 1.6, therefore resulting in a less robust controller. For the integrative case, Figure 8 shows the obtained T0 for the same interval, L ∈ [0.1, 5], and the same M0s = 1.6. The

min J(T0) = min{[Ms0 − Ms(T0)]2 + [IAE0r − IAEr(T0)]2 T0

T0

+ [IAEd0 − IAEd(T0)]2 }

where all the simulation conditions of the PID case are the same as those done for FSP. The minimization of J(T0) has been done by using classical interior-point methods, and it was solved through the Matlab Optimization Toolbox.24 To obtain a general rule for this tuning, first, the models are normalized considering a unitary gain (note that the process gain does not affect the result because the gain of the ideal and approximated controllers is the same) and a delay related to the time constant (for the stable case). With these considerations, the normalized models are Ps(s) =

e −L ′ s , s+1

Pi(s) =

e−Ls , s

L′ = L / T

Moreover, in the simulations, L′(L) is varied between very small values (lag-dominant processes) and high values (deadtime dominant processes) to cover most of the possible situations found in practice. Because of the normalization, the obtained solution in the optimization procedure is also normalized with T. Therefore, T0′ = T0/T is the argument of J. The proposed procedure can be applied to any value of the desired Ms. However, here, a typical Ms = 1.6 is used, as recommended in ref 1. For the stable case, Figure 7 shows the obtained T0′ for the interval L′ ∈ [0.1, 5] for M0s = 1.6 for the two optimization cases. The solution given by the robustness optimization procedure is shown as circles, and the solution of the robustness-performance optimization is shown as solid dots. In addition, two simple straight lines are used to approximate the set of points to derive a simple tuning rule. The final solution can be put then as a function T′0(L′) as follows: • For robustness: T0′ = 0.6L′ + 0.25 • For performance robustness trade-off: T′0 = 0.42L′ + 0.35 Remarks: (i) This procedure can be repeated for other values of M0s , obtaining similar curves; (ii) for the stable case study in

Figure 8. T0 tuning rule for the integrative case with Ms = 1.6 for the robustness criterion (circles) and for the performance-robustness trade-off (solid dots).

solutions given by the two optimization procedures are shown using the same type of points as in the stable case. Again, two simple straight lines are used to approximate the set of points to derive a simple tuning rule T0(L): • For robustness: T0 = 2.65L • For performance−robustness trade-off: T0 = 2.26L + 0.14 Note that for the integrative case study in FSP for FOPDT, IPDT, and UFOPDT Processes, where L = 1, T0 = 2.65 for 16817



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• Tune T0 ∈ [T0min, T0max] in the PID to minimize the following criterion:

robustness and T0 = 2.4 for performance should be used for Ms = 1.6, according to the results in Figure 8. As in that example, T0 = 2.75 was selected and the obtained PID controller has an Ms < 1.6, therefore giving a more robust controller. Tuning for the Unstable Case. In this case, the normalized plant model is given by Pu(s) =

e −L ′ s , s−1

min J(T0) = min{[IAE0r − IAEr(T0)]2 T0

T0

+ [IAEd0 − IAEd(T0)]2 }

which gives a trade-off between robustness and performance. Of course, in the PID simulations, the same conditions as in the FSP simulations are considered. Figure 9 shows the obtained optimal T0 (shown by solid points) for the interval L ∈ [0.1, 1.2]. Furthermore, in this

L′ = L / T

and again, the IAE is used to analyze the closed-loop performance. The unstable case has some particular properties, and thus, a different analysis should be done. First, note that to control an unstable process with dead time, a minimal feedback action is necessary to maintain nominal closed-loop stability, and therefore, detuning the controller (by increasing T0) has a limit (T0max). On the other hand, because of the delay, a minimal value of T0 (T0min) exists such that for smaller values, there is no guarantee of nominal stability (it is not possible to arbitrarily accelerate the closed-loop response). Moreover, these two conditions allow one to find an interval [T0min, T0max] where the optimal value of T0 should be found. This interval depends on the relationship L/T, and it becomes very narrow when L/T > 1. In fact, by making several simulation analysis, it was found that for L/T > 1.2, the possible solutions for T0 give such poor robustness that the closed-loop system can become unstable for very small errors of the delay estimation. Therefore, because in practice it is impractical to consider controllers for perfect modeled plants, the analysis of the T0 tuning was restricted to the interval L/T ∈ [0.1, 1.2]. Because of the particular properties of this case, the use of Ms as a robustness measurement was not found appropriate. Note that most of PID tuning rules proposed for unstable dead-time plants have very high values of Ms (see, for instance, the results in the unstable example presented in FSP for FOPDT, IPDT, and UFOPDT Processes). Therefore, a different robust condition for the PID is proposed using the following procedure: • Consider Pu and a delay estimation error of ΔL = 0.1 (10%) to compute the multiplicative modeling error dP(jω) = |e−0.1jω − 1|. • Initially, tune T0 in the PID to achieve the robust stability condition RIPID(ω0, T0) = dP(jω0). From this condition, it can be observed that ω0 ∈ [ωmin,ωmax], where ωmin = π/30ΔL, ωmax = 30π/ΔL, which is a frequency interval centered in ω = π/ΔL = 10π, where the peak value of dP(jω) occurs. This tuning is, in fact, the tuning for a PID with a delay margin of 10%. • To include a certain margin of robustness in the previous tuning, the condition is modified to find T0 such that RIPID(ω, T0) is tangent to εδP(jω) at one point in the interval [ωmin, ωmax], where ε = 1.5. This rule gives reasonably closed-loop responses for the considered uncertainty system. Now, to consider performance specifications by means of the IAE value, a procedure similar to the one used in the stable and integrative cases is proposed: • Tune T0 in the FSP controller for the robustness condition by using the tangent condition between RIFSP(ω, T0) and εdP(jω). • With the obtained T0 value, compute the IAEr and IAEd for the FSP controller and define as optimal indexes, IAE0r and IAE0d.

Figure 9. T0 tuning rule for the unstable case.

figure, the minimal value of T0 for nominal stability, the obtained T0 values for the robust condition, and the T0 of the robust FSP are shown for comparison purposes. Note that the PID tuning and FSP tuning give similar values for small L/T, which means that the PID can achieve reasonable results in these cases; however, this is not the case for bigger values of L/ T. On the other hand, the final values of T0 are smaller than the ones obtained for the robust condition only, because the optimizer tries to approximate the PID and FSP responses and the price to pay is a lower robustness. To complete the results, an analytical approximated expression is derived for the tuning rule: T0′ = 0.1342e 2.382L

which is plotted in the figure using a dotted line. The proposed rule can be analyzed using the unstable case study in FSP for FOPDT, IPDT, and UFOPDT Processes, where L = 0.4, and T0 = 0.5 should be used according to the results in Figure 9. Because in that example T0 = 0.4 was selected, the PID controller of example 3 is a less robust controller.

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CONCLUSIONS This paper has presented a unified PID design for FOPDT, IPDT, and UFOPDT processes. The proposed controller is simple to analyze and tune and it is based on a low-frequency approximation of the ideal dead-time compensation structure of the FSP. Given the process model, only one parameter is used to obtain a compromise between performance and robustness. 16818



Article

(9) Tyreus, B. D.; Luyben, W. L. Tuning PI controllers for integrator/dead time processes. Trans. ASME 1992, 31, 2628−2631. (10) Luyben, W. L. Tuning Proportional-Integral-Derivative Controllers for Integrator/Deadtime Processes. Ind. Eng. Chem. Res. 1996, 35, 3480−3483. (11) Depaor, A. M.; O’Malley, M. Controllers of Ziegler−Nichols type for unstable processes. Int. J. Control 1989, 49, 1273−1284. (12) Venkatashankar, V.; Chidambaram, M. Design of P and PI controllers for unstable FOPTD model. Int. J. Control 1994, 60, 137− 144. (13) Marchetti, G.; Scali, C.; Lewin, D. R. Identification and control of open loop unstable processes by relay methods. Automatica 2001, 37, 2049−2055. (14) Visioli, A. Optimal tuning of PID controllers for integral and unstable processes. IEE Proc.: Control Theory Appl. 2001, 148, 180− 184. (15) Ziegler, J.; Nichols, N. Optimum Settings for Automatic Controllers. Trans. ASME 1942, 64, 759−768. (16) Åström, K. J.; Hägglund, T. Revisiting the Ziegler−Nichols step response method for PID control. J. Process Control 2004, 14, 635− 650. (17) Lee, Y.; Lee, M.; Park, S.; Brosilow, C. PID controller tuning for desired closed-loop responses for SISO systems. AIChE J. 1998, 44, 106−116. (18) Padma, R.; Srinivas, M. N.; Chidambaram, M. A simple method of tuning PID controllers for stable and unstable FOPTD systems. Comput. Chem. Eng. 2004, 28, 2201−2218. (19) Normey-Rico, J. E.; Camacho, E. F. Control of Dead-time Processes; Spinger: Berlin, 2007. (20) Normey-Rico, J. E.; Camacho, E. F. Unified approach for robust dead-time compensator design. J. Process Control 2009, 19, 38−47. (21) Normey-Rico, J. E.; Camacho, E. F. Simple Robust Dead-Time Compensator for First-Order Plus Dead-Time Unstable Processes. Ind. Eng. Chem. Res. 2008, 47, 4784−4790. (22) Lee, Y.; Lee, J.; Park, S. PID controller tuning for integrating and unstable processes with time delay. Comput. Eng. Sci. 2000, 55, 3481−3493. (23) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (24) The Mathworks. Matlab Optimization Toolbox User’s Guide; http://www.mathworks.co.uk/access/helpdesk/help/pdf_doc/optim/ optim_tb.pdf, 2007.

Three case studies were presented to illustrate the tuning procedure and to show that the proposed controller allows for a closed-loop system with similar or better performance and robustness than other algorithms presented in the literature. However, the main contribution with respect to other existing methods is that the proposed approach deals with stable, integrative, and unstable processes using the same unified procedure. Moreover, a tuning procedure to select the free parameter considering a trade-off between robustness and performance is developed. This tuning also illustrates how far the PID performance is from the ideal solution given by the FSP. Furthermore, the presented method fulfills most of the recommendations proposed by Skogestad in ref 3mainly those dealing with well motivated rules, model-based and analytically derivedand those others saying that the rules work well on a wide range of processes. However, Skogestad also suggests that rules must be simple and easy to memorize. As discussed through the paper, the rules presented here are easy and simple, but in some cases, they are not too easy to memorize. Thus, future works will be focused on simplifications of the proposed tuning rules such that they are easy to memorize, for example, to analyze loop-shaping methods through optimization procedures to fit the frequency response of the approximated Cpid(s) controller.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes ⊥

A preliminary version of this work was published at IFAC Conference on Advances in PID Control PID’12, Brescia, Italy, 2012. The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been funded in part by the following projects: PHB2009-0008 (financed by the Spanish Ministry of Education; CNPq-BRASIL; CAPES-DGU 220/2010), DPI2011-27818-C02-01 (financed by the Spanish Ministry of Science and Innovation and EU-ERDF funds), and Controlcrop PIO-TEP-6174 (financed by the Consejeriá de ́ Economia,́ Innovación y Ciencia de la Junta de Andalucia).

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REFERENCES

(1) Aström, K. J.; Hägglund, T. Advanced PID Control; ISA − The Instrumentation, Systems and Automation Society: Research Triangle Park, NC 27709, 2005. (2) PID’12 IFAC Conference on Advances in PID Control; Brescia, Italy, March 2012. (3) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 92−102. (4) Grimholt, C.; Skogestad, S. Optimal PI-Control and Verification of the SIMC Tuning Rule; In ăIFAC Conference on Advances in PID Control PID’12: Brescia, Italy, March 2012. (5) O’Dwyer, A. Handbook of PI and PID Controller Tuning Rules; Imperial College Press: London, 2003. (6) Cohen, G. H.; Coon, G. A. Theoretical investigation of retarded control. Trans. ASME 1953, 75, 827−834. (7) Rivera, D.; Morari, M.; Skogestad, S. Internal model control 4. PID controller design. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 252−265. (8) Wang, Q. C.; Hang, C. C.; Yang, X. P. Single-loop controller design via IMC principles. Automatica 2001, 37, 2041−2048. 16819


Unified PID Tuning Approach for Stable, Integrative, and Unstable

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