Performance Assessment and Retuning of PID Controllers - American

Jan 16, 2009 - Proportional-integral-derivative (PID) controllers are un- doubtedly the most widely adopted controllers in industry owing to the advan...
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Ind. Eng. Chem. Res. 2009, 48, 2616–2623

Performance Assessment and Retuning of PID Controllers Massimiliano Veronesi and Antonio Visioli* Dipartimento di Elettronica per l’Automazione UniVersity of Brescia, Via Branze 38, I-25123 Brescia, Italy

In this article, we propose an algorithm for the set-point-following performance assessment of a PID controller and for the retuning of the parameters in case the obtained response is not satisfactory. The technique is based on the SIMC tuning rules and can be applied simply by evaluating a set-point step response. Simulation and experimental results obtained with laboratory-scale equipment show the effectiveness of the methodology. 1. Introduction Proportional-integral-derivative (PID) controllers are undoubtedly the most widely adopted controllers in industry owing to the advantageous cost/benefit ratios they are able to provide. To help the operator select the controller gains to address given control specifications, many tuning formulas have been devised,1 and autotuning functionalities are almost always available in commercial products.2,3 However, it is also recognized that, in many practical cases, PID controllers are poorly tuned because of the lack of time and the lack of skill of the operator. Actually, because there are hundreds of control loops in large plants, it is almost impossible for operators to monitor each of them manually. For these reasons, it is important to have automatic tools that are able to assess the performance of a control system and, if it is not satisfactory, to suggest a way to solve the problem (for example, if a bad controller tuning is detected, then new appropriate values of controller parameters are determined). Many performance assessment methodologies have been proposed in the literature and successfully applied in industrial settings.4 In general, although the proposed techniques can be viewed under the same framework (see ref 5 and references therein), they are generally divided into two categories:6 stochastic performance monitoring, in which the capability of the control system to cope with stochastic disturbances is of main concern (works that fall in this class mainly rely on the concept of minimum variance control7), and deterministic performance monitoring, in which performances related to more traditional design specifications such as set-point and loaddisturbance-rejection step-response parameters are taken into account.8 When deterministic requirements are considered, it is realized that unsatisfactory performance can be caused by different factors.9 Thus, there is a need to integrate different techniques, each of them designed to deal with a particular situation. Restricting the analysis to the tuning assessment of PID controllers, an iterative solution method for the determination of the minimum-variance PID controller was proposed in ref 10. Regarding deterministic performance monitoring, the achievable optimal performance in terms of the integrated absolute error for the set-point response was investigated in ref 11. Its knowledge could be exploited to evaluate the performance of an employed PI or PID controller, but once the tuning was recognized as unsatisfactory, a procedure for selecting new suitable values of the PID parameters was not given. In ref 12, the set-point-following performance of a PI controller was * To whom correspondence should be addressed. Tel.: +39-0303715460. Fax: +39-030-380014. E-mail: [email protected].

assessed by taking into account the step response that could be obtained by selecting the controller parameters by means of the internal model control (IMC) tuning rule, but again, a procedure on how to retune the controller was not given. On the contrary, this was presented in ref 13, but the overall technique (which is applicable only to a PI controller) requires a special experiment, which is not desirable in practical cases where the use of routine data is obviously preferred. Regarding loaddisturbance-rejection performance, a methodology to detect sluggish control loops was presented in ref 14 and further discussed in ref 15. It was also exploited in ref 16, where the technique presented was used to assess the tuning of a PI controller and then provide guidelines on how to retune it if necessary. A comprehensive description of all of these techniques is available in ref 17. It is clear that there is still the need for a methodology, based on routine data, that assesses the performance of a PID controller and provides a new tuning of all of the controller parameters. In this article, we focus on set-point-following performance, and we propose a methodology that evaluates a closed-loop step response; assesses the performance of the PID controller; and, if it is not satisfactory, provides a new tuning of the parameters. The technique aims at achieving the same performance as the SIMC (Skogestad IMC) tuning rules18 and is based on the determination of the sum of the time constants and the dead time of the process. It is worth noting that, in contrast to the approach of detecting poor performance and, as a consequence, using the automatic tuning functionality to determine a new set of PID parameters, here, the set-point response data are employed directly to determine the new values of the PID parameters. Thus, an automatic tuning experiment is not necessary, which is of great relevance especially if the experiment would be time- and energy-consuming and would influence the process routine operations. This article is organized as follows: The problem is formulated in section 2, where the SIMC tuning rule is also reviewed briefly. The performance assessment technique is described in section 3, and the algorithm for retuning the parameters is proposed in section 4 and discussed in section 5. Simulation results are given in section 6, and experimental results are presented in section 7. Conclusions are drawn in section 8.

Figure 1. Control scheme considered.

10.1021/ie800812b CCC: $40.75  2009 American Chemical Society Published on Web 01/16/2009

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2617

Figure 2. Results of example 1. Dash-dotted line: step response with initial parametervalues Kp ) 1.8, Ti ) 12, and Td ) 0. Solid line: step response after the application of the retuning algorithm. Dashed line: step response with the SIMC tuning rule.

Figure 3. Results of example 1. Dash-dotted line: step response with initial parameter values Kp ) 1, Ti ) 15, and Td ) 2. Solid line: step response after the application of the retuning algorithm. Dashed line: step response with the SIMC tuning rule.

2. Problem Formulation We consider the unity-feedback control system of Figure 1 where the process P is controlled by a PID controller whose transfer function is in series (“interacting”) form

(

C(s) ) Kp

)

Tis + 1 (Tds + 1) Tis

(1)

The series form was chosen for the sake of simplicity, as the SIMC tuning rule that will be employed in the methodology applies directly to this form. However, the use of other forms is straightforward through the application of suitable translation formulas to determine the values of the parameters.17 Note also that the use of a first-order filter that makes the controller transfer function proper was neglected for the

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Figure 4. Results of example 2. Dash-dotted line: step response with initial parameter values Kp ) 1, Ti ) 6, and Td ) 0. Solid line: step response after the application of the retuning algorithm. Dashed line: step response with the SIMC tuning rule.

Figure 5. Results of example 2. Dash-dotted line: step response with initial parameter values Kp ) 0.4, Ti ) 10, and Td ) 1. Solid line: step response after the application of the retuning algorithm. Dashed line: step response with the SIMC tuning rule.

sake of clarity, but it can easily be selected so that it does not influence the PID controller dynamics significantly. The aim of the proposed methodology is to evaluate the closed-loop system response when a step signal is applied to the set point and to assess the tuning of the PID controller. For the sake of simplicity, without loss of generality, we consider that the set-point step signal is applied starting from null initial conditions. To assess the control performance, a

benchmark that represents the desired performance has to be selected so that the current control performance can be evaluated against it. Here, for this purpose, we employ the set-point-following performance achieved by the PID controller designed according to the method proposed in ref 18, which has been shown to be very effective for a wide range of processes with an overdamped step response and which is reviewed briefly hereafter. This method consists of using

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where it is suggested to select the desired closed-loop time constant as the dead time of the process. It results that the PID parameters in eq 1 are selected as (SIMC tuning rule) Kp )

τ1 , Ti ) τ1, Td ) τ2 2µθ

(5)

Define as T0 the sum of the time constants and of the dead time of the process, i.e.

∑τ

T0 ≡

i0

+ θ0

(6)

i

Then, as a consequence of eqs 4 and 5, it is trivial to verify that T0 ) τ1 + τ2 + θ ) Ti + Td +

Ti 2µKp

(7)

Note that the SIMC tuning rule aims at achieving a closedloop transfer function (this can easily be ascertained by again approximating the delay term as e-θs ≈ 1 - θs) F(s) ≡

C(s) P˜(s) 1 e-θs ≈ ˜ θs +1 1 + C(s) P(s)

(8)

for which the resulting step-response 2% settling time is Ts2 ≈ 5θ

(9)

and the step-response integrated absolute error (IAE) is Figure 6. Experimental setup for the level control experiment (only one tank was used).

a given tuning rule based on a process model reduced by applying the so-called “half rule”, which states that the largest neglected (denominator) time constant is distributed evenly to the effective dead time and the smallest retained time constant. In practice, the following (possibly high-order) process transfer function with real poles is considered (note that, in the retuning algorithm proposed in this work, the a priori knowledge of the process model is not required) P(s) )

∏

µ (τi0s + 1)

e-θ0s

(2)

i

where the time constants are ordered according to their magnitude (that is, τ10 > τ20 >...). Then, a second-order-plusdead-time (SOPDT) transfer function P˜(s) )

µ e-θs (τ1s + 1)(τ2s + 1)

(3)

is obtained by setting τ1 ) τ10, τ2 ) τ20 +

τ30 τ30 , θ ) θ0 + + 2 2

∑τ

i0

IAE )

∞ 0 |e(t)|

dt ) 2Aθ

(10)

where e(t) ) r(t) - y(t) and A is the amplitude of the set-point step. 3. Performance Assessment To assess the performance of the PID controller in the setpoint-following task, it is therefore necessary to verify, in principle, that eqs 7 and 9 (or, alternatively, eq 10) are verified. The determination of the 2% settling time, Ts2, or the integrated absolute error, IAE, can be performed easily by evaluating the step response of the closed-loop system. Note that the use of the integrated absolute error is preferable because, being based on the integral of a signal, it is more robust to the measurement noise. Then, the apparent dead time, θm, of the system can be evaluated by considering the time interval from the application of the step signal to the set point and the time instant when the process output attains 2% of the new set-point value A, that is, when the condition y > 0.02A occurs. Actually, from a practical point of view, to cope with the measurement noise, a simple sensible solution is to define a noise band, NB19 (whose amplitude should be equal to the amplitude of the measurement noise), and to rewrite the condition as y > NB. Then, eqs 9 and 10 can be rewritten, respectively, as

(4)

Ts2 ≈ 5θm

ig4

It is worth noting that, unlike in ref 18, the presence of positive zeros is not considered in eq 2. However, the associated time constants can be simply added to the dead time of the process.18 Once the SOPDT process model is obtained, the PID parameters are determined by applying the IMC design procedure (and by approximating the delay term as e-θs ≈ 1 - θs),

∫

(11)

and IAE )

∫

∞

0

|e(t)| dt ) 2Aθm

(12)

With respect to eq 7, it is necessary to determine the process gain, µ, and the sum of the lags and of the dead time of the process, T0. The process gain, µ, can be determined by

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Figure 7. Experimental step response obtained with the initial PID parameters Kp ) 1.5, Ti ) 7, and Td ) 0.

considering the following trivial relations involving the final steady-state value of the control variable u lim u(t) )

tf+∞

Kp Ti

∫

∞

0

e(t) dt )

A µ

(13)

By applying the final value theorem to the integral of eu when a step is applied to the set-point signal, we finally obtain

lim

tf+∞

∫ e (V) dV t

0

u

and therefore, we have ) lim s

Ti

µ)A Kp

∫

∞

sf0

(14)

µKpc˜(s)[q(s)-e-θ0s] A s [T sq(s) + µK c˜(s) - e-θ0s]s

e(t) dt

0

) A lim

The determination of the sum of the lags and the dead time of the process can be performed by considering the variable eu(t) ) µu(t) - y(t)

(15)

sf0

(

i

[

p

q(s) - 1 1 - e-θ0s + s s

) A θ0 +

∑τ

]

)

i0

i

) AT0

(20)

By applying the Laplace transform to eq 15 and by expressing u and y in terms of r, we have Eu(s) ) µU(s) - Y(s) )

C(s)[µ - P(s)] R(s) 1 + C(s) P(s)

(16)

At this point, for the sake of clarity, it is convenient to write the controller and process transfer functions, respectively, as C(s) )

Kp c˜(s) Tis

where c˜(s) ≡ (Tis + 1)(Tds + 1) (17)

and µe-θ0s P(s) ) q(s)

where q(s) ≡

∏ (τ

i0s

+ 1)

(18)

i

Thus, the sum of the lags and the dead time of the process can be obtained by evaluating the integral of eu(t) at the steady state (which does not depend on the PID parameters) when a step signal is applied to the set point and by dividing it by the amplitude A of the step. It is worth noting that the values of both the gain and the sum of the lags and the dead time of the process are determined by considering the integrals of signals, and therefore, the method is inherently robust to the measurement noise. For the purpose of assessing the controller performance based on eqs 7 and 10, it is worth considering the sigma index (SI) and closed-loop index (CI) performance indexes

Then, eq 16 can be rewritten as Eu(s) )

µKpc˜(s) -θ0s

Tisq(s) + µKpc˜(s) e

[q(s) - e-θ0s]R(s) (19)

Ti + Td + SI )

T0

Ti 2µKp

(21)

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Figure 8. Experimental step response obtained with the retuned PID (Kp ) 1.5, Ti ) 18.2, and Td ) 5.27).

CI )

2Aθm

∫

∞

0

(22)

|e(t)| dt

In principle, the performance obtained by the control system is considered to be satisfactory if CI ) 1. From a practical point of view, however, the controller is considered to be well-tuned if CI > CId with CId ) 0.6. Note that this last value was selected by considering the SIMC tuning rule applied to many different processes,18 but in any case, another value of CId can be selected by the user depending on how tight the control specifications are. If the controller turns out to be badly tuned, then two cases might occur. If the value of SI is less than 1, this means that the current tuning is based on an underestimation of the process lags and/or of the dead time. Thus, an improvement of the performance can be achieved by decreasing the value of Kp and/ or by increasing the value of Ti and/or Td (see eq 7). On the contrary, if the value of SI is greater than 1, this means that the current tuning is based on an overestimation of the process lags and/or of the dead time. Thus, an improvement of the performance can be achieved by increasing the value of Kp and/or by decreasing the value of Ti and/or Td. These considerations are exploited in the methodology for the retuning of the PID controller proposed in the next section. Remark 1: It is worth noting that selecting the closed-loop index, CI, as a performance index implies that the desired integrated absolute error of the step response is 2Aθ (see eq 12). Actually, minimizing the integrated absolute error is meaningful because this yields, in general, a low overshoot and a low settling time at the same time.20 Then, although the minimum integrated absolute error that can be achieved for a single-loop system (with a general feeback controller) is 1.31Aθ,11 the desired value of 2Aθ is sensible if the fixed structure of the controller, the robustness issue, and the control effort are taken into account.

4. Retuning Algorithm If the performance provided by the controller results to be unsatisfactory, the PID controller needs to be retuned. Here, we propose a method that aims at tuning the PID controller according to the SIMC tuning rules (without knowing the process model a priori). For this purpose, we exploit the following considerations: The closed-loop step response has to be evaluated and θm and T0 have to be calculated according to the technique described in section 3 (see eq 20). Then, if an overshoot occurs, the proportional gain Kp is decreased, and the integral time constant Ti and the derivative time constant Td are subsequently calculated by means of eqs 5 and 7. If the resulting value of Ti is less than the value of Td, then Ti has to be increased, and the values of Kp and Td are updated again according to eqs 5 and 7. The procedure is iterated until a value Td > 0 is obtained. Conversely, if the closed-loop step response is underdamped (namely, there is no overshoot), the procedure starts by increasing the proportional gain, Kp, and then the same steps of the case with overshoot apply. By taking into account the considerations made in section 3, a sound measure on how much initially increasing or decreasing the parameters is given by the sigma index SI. Formally, starting from given values of Kp, Ti, and Td, the algorithm for retuning the PID controller is described as follows: Retuning Algorithm. (1) Evaluate a closed-loop step response y(t) and the corresponding control variable u(t) and determine µ, θm, and T0 (see eqs 14 and 20). (2) Calculate SI according to eq 21. (3) Set Kpi ) Kp (store the initial value of the proportional gain). (4) If maxt y(t) > A (i.e., if there is overshoot), then (a) Set Kp ) Kpi × min{SI, 1/SI}. (b) Set Ti ) 2µKpθm. (c) Set Td ) T0 - Ti - Ti/2µKp. (d) If Ti < Td, then (i) Set Ti ) Ti × max{SI, 1/SI}.

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(ii) Set Kp ) Ti/2µθm. (iii) Set Td ) T0 - Ti - Ti/2µKp. (e) If Td < 0, then (i) If SI > 1, then set SI ) 2SI; else set SI ) SI/2. (ii) Go to step 4a. (5) If maxt y(t) e A (i.e., if there is no overshoot), then (a) Set Kp ) Kpi × max{SI, 1/SI}. (b) Set Ti ) 2µKpθm. (c) Set Td ) T0 - Ti - Ti/2µKp. (d) If Ti < Td, then (i) Set Ti ) Ti × max {SI, 1/SI}. (ii) Set Kp ) Ti/2µθm. (iii) Set Td ) T0 - Ti - Ti/2µKp. (e) If Td < 0, then (i) If SI < 1, then set SI ) 2SI; else set SI ) SI/2. (ii) Go to step 5a. (6) End. 5. Discussion The proposed retuning algorithm, although justified by rigorous motivations, is clearly empirical, and therefore, it does not generally achieve exactly the same results as the SIMC tuning rules. However, it has the clear advantage that it does not require any intervention by the operator and, therefore, can be implemented in an automatic tuning framework. From a practical point of view, to cope with measurement noise, the condition maxt y(t) > A that determines whether an overshoot has occurred in the closed-loop step response can be conveniently replaced by maxt y(t) > (A + NB), where NB is the already mentioned noise band. In any case, it is worth stressing that the devised methodology, in addition to being mainly based on integral indexes and, therefore, being inherently robust to measurement noise, can be applied off-line, so that any (possibly noncausal) standard filtering technique can be employed to filter the data before the application of the algorithm. It is also worth stressing that the value of T0 is obtained independently of the values of the PID parameters. This is an advantage with respect to the use of another method for the identification of the process transfer function, whose result depends on the control variable and process variable signals. Further, it is not guaranteed that, if another identification method (for example, the standard least-squares approach21) were employed, the resulting transfer function would have only real poles so that the SIMC tuning rules could be applied17 (see section 6). This kind of problem also generally occurs with other tuning rules. The result of the retuning algorithm actually depends on the initial tuning of the parameters, and the algorithm was devised in such a way that the proportional gain is modified first, followed by the integral time constant, and then the derivative time constant. The conditions at steps 4d, 4e, 5d, and 5e are evidently necessary to avoid obtaining final parameter values that do not have a sensible physical meaning. Note that there is no need to repeat the methodology because, in further iterations, the sigma index will be equal to 1. 6. Simulation Results In all of the following simulation examples, we set A ) 1, and there was an absence of noise, so that the apparent dead time of the process was determined when the process output attained 2% of its final value. Initial values of the PID parameters were selected arbitrarily in order to present different

situations (underdamped and overdamped responses). Further, a first-order filter that made the controller transfer function proper was employed. Its time constant was selected as Td/10. 6.1. Example 1. As a first example, we considered the process P1(s) )

1 e-s (s + 1)3

(23)

As a first experiment, we initially set Kp ) 1.8, Ti ) 12, and Td ) 0. The obtained step response is shown in Figure 2 (dashdotted line). We obtained CI ) 0.35 (with a 2% settling time of Ts2 ) 55.12), which clearly indicated that the controller needed to be retuned (as is clear upon looking at the step response). Then, the retuning algorithm was applied, starting from the obtained values of µ ) 1, θm ) 1.45, T0 ) 4, and SI ) 3.83. Because the response clearly exhibited an overshoot, step 4 of the retuning algorithm played the key role. Finally, we obtained a PID controller with Kp ) 0.47, Ti ) 1.37, and Td ) 1.18, which yielded the step response shown in Figure 2 as a solid line. The corresponding value of CI was 0.88, and the 2% settling time was Ts2 ) 9.06. To better evaluate the achieved performance, the response obtained by using the SIMC tuning rule (which yielded Kp ) 0.33, Ti ) 1, and Td ) 1.5; note that, in this case, the model of the process was assumed to be known) is also shown (dashed line). In the latter case, the values are CI ) 0.85 and Ts2 ) 9.45. From the results, it appears that the retuning algorithm allowed the performance to be improved significantly, even though the initial tuning provides a very oscillatory response. As a second experiment, we initially set Kp ) 1, Ti ) 15, and Td ) 2. In this case, the obtained step response was underdamped (see Figure 3), with CI ) 0.16 and Ts2 ) 93.3. The retuning algorithm, in which step 6 played the key role, was applied starting from the obtained values of µ ) 1, θm ) 1.17, T0 ) 4, and SI ) 0.77. The resulting PID parameters were Kp ) 0.77, Ti ) 1.80, and Td ) 1.03, and the corresponding step response is shown in Figure 3 as a solid line. The resulting value of CI was 0.74, and the 2% settling time was Ts2 ) 11.15. The step response obtained with the SIMC tuning rules is shown again for the sake of comparison. It can be seen that, also in this case, the performance was improved significantly, and although the result obtained was slightly worse than in the previous case, the practical relevance of the methodology is evident. Remark 2: It is worth noting that application of a least-squares method for the identification of a second-order-plus-dead-time process transfer function would result in a transfer function with complex conjugate poles in both cases. 6.2. Example 2. As a second example, we considered the high-order process P2(s) )

1 (s + 1)8

(24)

In this case, we initially set Kp ) 1, Ti ) 6, and Td ) 0, and these parameters yields the step response shown in Figure 4 (dash-dotted line). We obtained CI ) 0.46 (with a 2% settling time of Ts2 ) 73.28), which indicated that the retuning algorithm had to be applied. Starting from the obtained values of µ ) 1, θm ) 3.26, T0 ) 8, and SI ) 2.25, we eventually obtained a PID controller with Kp ) 0.44, Ti ) 2.90, and Td ) 1.85, which yielded the step response shown in Figure 4 as a solid line. The corresponding value of CI was 0.83, and the 2% settling time was Ts2 ) 26.88. The response obtained by using the SIMC

Ind. Eng. Chem. Res., Vol. 48, No. 5, 2009 2623

tuning rule (for which Kp ) 0.09, Ti ) 1, and Td ) 1.5) is also shown. In this latter case, the values are CI ) 0.54 and Ts2 ) 36.37. It appears that, also for this high-order process, the retuning algorithm allows the performance to be improved significantly. When a different set of parameters was selected initially, namely, Kp ) 0.4, Ti ) 10, and Td ) 1, the resulting step response was the one shown in Figure 5, for which CI ) 0.26 and Ts2 ) 109.1. By applying the retuning algorithm with θm ) 3.26, T0 ) 8, and SI ) 1.47, we eventually obtained a PID controller with Kp ) 0.59, Ti ) 3.83, and Td ) 0.91, which yielded the step response shown in Figure 5 as a solid line. The resulting value of CI was 0.75, and the 2% settling time was Ts2 ) 29.36. Again, the step response obtained with the SIMC tuning rule is shown for comparison. The effectiveness of the proposed methodology emerges in this case as well. 7. Experimental Results To verify the effectiveness of the devised technique in practical applications, a laboratory experimental setup (made by KentRidge Instruments) was used (see Figure 6). Specifically, the apparatus consisted of small perspex tower-type tank (with a cross-sectional area of 40 cm2) in which a level control was implemented by means of a PC-based controller. The tank was filled with water by means of a pump whose speed was set by a dc voltage (the manipulated variable), in the range of 0-5 V, through a pulse-width-modulation (PWM) circuit. The tank was fitted with an outlet at the base in order for the water to return to a reservoir. The level of the water was measured by a capacitive-type probe that provided an output signal between 0 (empty tank) and 5 V (full tank). For the sake of simplicity, in the following discussion, the level variable is expressed in volts. Note that the system exhibits nonlinear dynamics, because the flow rate out of the tank depends on the square root of the level. The control task was to achieve a process variable transition from 2 to 3 V (thus, A ) 1). Initially, the following tuning was selected: Kp ) 1.5, Ti ) 7, and Td ) 0, and the obtained results are shown in Figure 7. The value of the closed-loop index CI was 0.38 (Ts2 ) 120 s). Hence, the retuning algorithm was applied, using θm ) 3.0, T0 ) 26.47, and SI ) 0.31. The PID parameters resulting from the retuning algorithm were Kp ) 1.5, Ti ) 18.2, and Td ) 5.27, and the corresponding step response is shown in Figure 8 (note the different time scale with respect to Figure 7). The resulting closed-loop index was CI ) 0.70, and the settling time was reduced significantly to Ts2 ) 24.5 s. Thus, the proposed methodology is clearly effective in a practical context. 8. Conclusions A methodology for the set-point-following performance assessment of a PID controller and for the retuning of the parameters has been proposed. The method considers the SIMC tuning rule (for the set-point-following task) as a benchmark, and it is capable of effectively tuning the PID controller,

independent of the initial controller parameters. Because the technique is based on the determination of the sum of the dead time and the lags of the process through the integration of appropriate signals, it is robust to measurement noise and suitable for implementation in a practical context, as was demonstrated by the presented examples. Acknowledgment This work was partially supported by MUR scientific research funds. Literature Cited (1) O’Dwyer, A. Handbook of PI and PID Controller Tuning Rules; Imperial College Press: London, 2006. (2) Leva, A.; Cox, C.; Ruano, A. Hands-on PID autotuning: A guide to better utilisation; IFAC Professional Brief; Amsterdam, 2001. (3) Åstro¨m, K. J.; Ha¨gglund, T. AdVanced PID Control; ISA Press: Research Triangle Park, NC, 2006. (4) Jelali, M. An overview of control performance assessment technology and industrial applications. Control Eng. Pract. 2006, 14, 441. (5) Huang, B.; Shah, S. L. Performance Assessment of Control Loops: Theory and Applications; Springer: London, 1999. (6) Qin, S. J. Control performance monitoringsA review and assessment. Comput. Chem. Eng. 1998, 23, 173. (7) Harris, T. J.; Seppala, C. T.; Desborough, L. D. A review of performance monitoring and assessment techniques for univariate and multivariate control systems. J. Process Control 1999, 9, 1. (8) Eriksson, P.-G.; Isaksson, A. J. Some aspects of control loop performance monitoring. In Proceedings of the IEEE International Conference on Control Applications; IEEE Press: Piscataway, NJ, 1994; p 1029. (9) Patwardhan, R. S.; Shah, S. L. Issues in performance diagnostic of model-based controllers. J. Process Control 2002, 12, 413. (10) Ko, B.-S.; Edgar, T. F. PID control performance assessment: The single-loop case. AIChE J. 2004, 50, 1211. (11) Huang, H.-P.; Jeng, J.-C. Monitoring and assessment of control performance for single loop systems. Ind. Eng. Chem. Res. 2002, 41, 1297. (12) Swanda, A. P.; Seborg, D. E. Controller performance assessment based on set-point response data. In Proceedings of the American Control Conference; IEEE Press: Piscataway, NJ, 1999; p 3863. (13) Thyagarajan, T.; Yu, C.-C. Improved autotuning using the shape factor from relay feedback. Ind. Eng. Chem. Res. 2003, 42, 4425. (14) Ha¨gglund, T. Automatic detection of sluggish control loops. Control Eng. Pract. 1999, 7, 1505. (15) Kuehl, P.; Horch, A. Detection of sluggish control loopssExperiences and improvements. Control Eng. Pract. 2005, 13, 1019. (16) Visioli, A. Method for proportional-integral controller tuning assessment. Ind. Eng. Chem. Res. 2006, 45, 2741. (17) Visioli, A. Practical PID Control; Springer: London, 2006. (18) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291. (19) Åstro¨m, K. J.; Ha¨gglund, T.; Hang, C. C.; Ho, W. K. Automatic tuning and adaptation for PID controllerssA survey. Control Eng. Pract. 1993, 1, 699. (20) Shiskey, F. G. Feedback Controllers for the Process Industries; McGraw-Hill: New York, 1994. (21) Sung, S.-W.; Lee, I.-B.; Lee, B.-K. On-line process identification and automatic tuning method for PID controllers. Chem. Eng. Sci. 1998, 53, 1847.

ReceiVed for reView May 21, 2008 ReVised manuscript receiVed November 4, 2008 Accepted December 9, 2008 IE800812B