Automatic Tuning of Feedforward Controllers for Disturbance Rejection

Article pubs.acs.org/IECR

Automatic Tuning of Feedforward Controllers for Disturbance Rejection Massimiliano Veronesi† and Antonio Visioli*,‡ †

Yokogawa Italy, Milan, Italy Dipartimento di Ingegneria Meccanica e Industriale, University of Brescia-Italy, Via Branze 38, I-25123 Brescia, Italy

‡

ABSTRACT: We propose a method for the automatic tuning of the feedforward compensator for proportional-integralderivative control loops in order to reject disturbances acting on the process. The parameters of the compensator are automatically computed after estimating the disturbance transfer function by using closed-loop routine operating data. Simulation and experimental results show the effectiveness of the methodology.

1. INTRODUCTION Proportional-integral-derivative (PID) controllers are the most employed controllers in industry owing to the cost/benefit ratio they are able to provide. In fact, despite their relative simplicity, they are capable of providing satisfactory performance for a wide range of processes. Further, many tuning rules1 have been devised, and automatic tuning methodologies are also available to make their design easier and therefore more suitable from an industrial point of view. It has also to be recognized that the great success of PID controllers is also due to the implementation of those additional functionalities (set-point weighting, antiwindup, etc.) that allow the user to improve the performance in practical cases.2 Among these additional functionalities, the use of a feedforward action plays a major role in improving the performance (when a sufficiently accurate model of the system can be estimated) both for the set-point following3 and for the load disturbance rejection task.4−6 In the latter case, which is the one considered in this paper, the disturbance must be measurable. For an overview of feedforward methodologies in the context of PID control, see ref 7 and the references therein contained. In this paper, which is an extended version of ref 8, we present an automatic tuning procedure for the automatic tuning of a feedforward block for the compensation of disturbances. With the same rationale employed in refs 9−12, the method consists of estimating the disturbance transfer function by evaluating a closed-loop disturbance step response (where a, possibly roughly tuned, controller is employed). Thus, there is the great advantage (compared to other more sophisticated, possibly frequency domain, identification methodologies that can be found in the literature) that it is therefore possible to use routine data without stopping the running control loop, which is obviously preferable in industrial settings. Further, the estimation procedure is based on very simple computations (so that it can be easily implemented in Distributed Control Systems) and on the integration of a signal, and it is therefore inherently robust to the measurement noise. On the basis of the obtained model of the disturbance, the feedforward block can then be designed by following a standard technique or a more recent one.13 The proposed methodology can be used together with the automatic tuning methodology for the feedback PID controller © 2014 American Chemical Society

proposed in ref 9 (which is briefly reviewed in section 3) in order to provide an overall control system automatic design. The paper is organized as follows. In section 2 the feedforward control scheme is described, as well as tuning techniques. In section 3 the automatic tuning methodology is described, by highlighting in particular the procedure for the estimation of the parameters, which plays a key role. Simulation results are given in section 4 while experimental results obtained with a laboratory setup are shown in section 5. Finally, conclusions are drawn in section 6.

2. FEEDFORWARD CONTROL SCHEME We consider the unity-feedback control system of Figure 1 where the self-regulating process P with an overdamped open-loop step response has the transfer function

Figure 1. The considered feedback control scheme.

P(s) =

k e−sL0 , q(s)

q(s) =

∏ (Tsj + 1),

T1 ≥ T2 ≥ ...

j

(1)

As an estimated model of the process (1), the following first-orderplus-dead-time (FOPDT) transfer function can be considered: P(̃ s) =

k e−sL Ts + 1

(2)

It is worth recalling that this simple model may come from a model reduction technique like the so-called “half-rule”14 which states that the largest neglected (denominator) time constant is Received: Revised: Accepted: Published: 2764

September 17, 2013 January 28, 2014 January 29, 2014 January 29, 2014 dx.doi.org/10.1021/ie403089f | Ind. Eng. Chem. Res. 2014, 53, 2764−2770

Industrial & Engineering Chemistry Research

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distributed evenly to the effective dead time and the smallest retained time constant. Thus, we have T T = T1 + 2 , 2

T L = L0 + 2 + 2

H (s ) = −

(10)

is added to the control scheme, which therefore is modified as shown in Figure 2. This means that, by assuming that the

∑ Tj (3)

j≥3

G̃ (s) P(̃ s)

Thus, the value of the sum of lags and delay of the model is the same one as that of the real process; that is, T0 := ∑ Tj + L0 = T + L (4)

j

Hence, T0 is a relevant process parameter that is worth estimating for the purpose of retuning the PID controller, as it will be shown in the following sections. Note also the presence of positive zeros is not considered explicitly in (1); however, the associated time constants can be simply added to the dead time of the process.14 The feedback controller is a PID controller whose transfer function is in series (“interacting”) form: ⎛ Ts + 1 ⎞⎛ Tds + 1 ⎞ ⎟⎟ C(s) = K p⎜ i ⎟⎜⎜ ⎝ Tsi ⎠⎝ Tf s + 1 ⎠

Figure 2. The control scheme with feedforward for disturbance rejection.

FOPDT transfer functions (2) and (8) are accurate models both for the process and for the disturbance, the following compensator can be added in the control strategy:16 μ Ts + 1 −sα H (s ) = − e (11) k τs + 1

(5)

where the dead time term α is chosen as

where Kp is the proportional gain, Ti is the integral time constant, and Td is the derivative time constant. The series form has been chosen for the sake of simplicity; however, the use of other forms is straightforward by suitably applying translation formulas to determine the values of the parameters.2 A first-order filter on the derivative action is also applied in order to reduce the actuator wear caused by the unavoidable process variable measurement noise. The filter time constant Tf is usually selected in order to filter the high-frequency noise and, at the same time, in order for the cutoff frequency to be higher than the system bandwidth15 (in the simulations of section 4 it will be chosen as Tf = Td/10). The controller transfer function can therefore be rewritten as C(s) =

Kp Tsi

c(s),

c(s) =

(Tsi + 1)(Tds + 1) Tf s + 1

α = max(0, θ − L)

as the causality of the feedforward block has to be ensured. Obviously, the ideal feedforward compensation (that is, the disturbance effect is completely rejected) is possible only if θ > L. On the other hand, when θ < L, the delay in H(s) is set equal to zero, and some effect on the process variable cannot be avoided. An alternative technique for the tuning of the feedforward block H when θ < L is that proposed in ref 13, where the role of the feedback controller is taken into account and the high frequency gain of the compensator is reduced in order to decrease the control effort. In this context, the feedforward block is selected as Ts + 1 −αs H̃ (s) = −k ff e Tps + 1

(6)

μ e−sθ0 , g (s )

g (s ) =

∏ (1 + sτj),

⎧τ if L ≤ θ ⎪ ⎪ L−θ if 0 < L − θ ≤ 1.7(τ − θ) Tp = ⎨ τ − 1.7 ⎪ ⎪0 if L − θ > 1.7(τ − θ ) ⎩

τ1 ≥ τ2 ≥ ...

j

(7)

⎧μ Kp ⎪ − μ(Tp − τ ) if θ ≥ L Ti ⎪k k ff = ⎨ Kp ⎪μ μ(Tp − τ − θ + L) if θ < L ⎪ − Ti ⎩k

(8)

where the value of the sum of lags and delay of the model is denoted as τ0 =

(15)

3. AUTOMATIC TUNING The proposed automatic tuning methodology consists of first estimating the process and disturbance transfer function and then tuning the feedforward controller by means of one of the techniques described in the previous section. The parameters of the process transfer function (2) can be obtained by applying the method already proposed and discussed in ref 9 and also applied in an industrial application as reported in ref 10.

∑ τj + θ0 = τ + θ j

(14)

and

which, by applying again the “half-rule”, can be eventually reduced to a FOPDT transfer function μ e−sθ G̃(s) = τs + 1

(13)

where

Finally, by following a reasoning similar to that applied for the process model, the (measurable) process disturbance can be represented by the following model: G (s ) =

(12)

(9)

It is worth noting that the classical load disturbance modeling, where the disturbance is added to the control variable, is a specific case of the scheme of Figure 1 with G = P. To compensate effectively for the disturbance, the classical ideal feedforward compensator 2765

dx.doi.org/10.1021/ie403089f | Ind. Eng. Chem. Res. 2014, 53, 2764−2770


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The determination of the dynamic parameters of G(s) is less trivial, although still based on the final value theorem. The following variable w is considered:

In particular, by evaluating a closed-loop response when a step signal of amplitude As is applied to the set-point r (and the PID controller has been previously tuned, possibly roughly), the sum of the lags and of the dead time of the process can be obtained by calculating the integral of v(t), defined as v(t ) = ku(t ) − y(t )

w = ku + μd − y kC(s)G(s) G (s ) d(s) + μ d(s) − d(s) 1 + C(s)P(s) 1 + C(s)P(s) −kC(s)G(s) + μ(1 + C(s)P(s)) − G(s) d(s) = 1 + C(s)P(s) ⎛ −μkK c(s)e−sθ0 sTq p i (s ) ⎜ = +μ −sL0 ⎜ sTg sTq i (s ) i (s) + kK pc(s)e ⎝

=−

(16)

where u is the manipulated variable and y is the process variable. In fact, it can be shown that9 T0 =

1 lim A s t →+∞

∫0

t

v (ξ ) d ξ

(17)

It appears that the steady-state value of v(t) does not depend on the PID parameters, but only on the process parameters (indeed, this is a nice feature of the method as it can be applied with any controller parameters). Further, by means of a simple application of the final value theorem, the process gain can be determined as

k = As

+

Kp ∫

0

e(t ) dt

∫ t →+∞ t

where e = r − y is the control error. Note that both the value of the gain and of sum of the lags and of the dead time of the process are determined by considering the integral of signals and therefore the method is inherently robust to the measurement noise. Finally, the apparent dead time L of the process can be evaluated by considering the time interval from the application of the step signal to the setpoint and the time instant when the process output attains 2% of the new set-point value As, namely, when the condition |r − y| > 0.02As occurs. Actually, from a practical point of view, in order to cope with the measurement noise, a simple sensible solution is to define a noise band NB17 (whose amplitude should be equal to the amplitude of the measurement noise) and to rewrite the condition as y > NB. Once the apparent dead time has been estimated, the time constant of the process can be simply calculated as

=

t →+∞

∫t

e(ξ) dξ = − 0

μTi Ad kK p

kK p

∫ AT t d i

0

w(ξ) dξ = lim s s→0

0

W (s) Ad s s

⎞ − q(s)e−sθ0 + g (s)e−sL0 Ad Ti ⎛ ⎜⎜lim μkK p + μ − μ⎟⎟ kK p ⎝s → 0 sTq ⎠ i (s)g (s) q(s) g (s)

⎞ ⎟ ⎟⎟ ⎠

∑ τj − ∑ Tj) = μAd(τ0 − T0) j

j

(23)

Thus, the sum of all the lags of G(s) can be obtained as 1 μAd

τ0 = T0 +

∫t

∞

w(ξ) dξ 0

(24)

Note that also in this case the estimation of the process parameters is based on the integral of signals and therefore the method is inherently robust to the measurement noise. Further, the process parameters are obtained independently on the values of the PID parameters, because the estimation is based on steady-state values of the variables. Then, as in the process parameters estimation procedure, the apparent dead time θ of the system can be evaluated by considering the time interval from the occurrence of the load disturbance and the time instant when the process output attains the 2% of the set-point value, namely, when the condition |y ̅ − y| > 0.02 μAd occurs, where y ̅ is the current steady state value of the process variable (the noise band concept can be employed in order to cope with measurement noise). Then, the time constant τ of transfer function (8) can be simply obtained as τ = τ0 − θ

(25)

A similar approach can be employed to estimate the disturbance transfer function also if the feedforward block is already in place (again, possibly roughly tuned). Denote as

(20)

∞

e(t ) dt

t

= μAd (θ0 − L0 +

Thus, the gain of G(s) can be determined easily as μ=−

(22)

q(s) ⎛ 1 − g(s) ⎞⎟ ⎜ e−s(L0 − θ0) − 1 = μAd lim⎜ + ⎟⎟ s → 0⎜ s s ⎠ ⎝

(19)

t

⎞ μe−sθ0 ⎟ d(s) g (s) ⎟⎠

⎛ e−s(L − θ0) − ⎜ = μAd lim⎜g (s)e−sθ0 s → 0⎜ s ⎝

On the basis of the determined process model, the PID controller can be retuned by applying anyone of the tuning rules proposed in the literature,1 depending on the control requirements. In the simulation results given in section 4, the tuning procedure proposed in ref 9 (based on the SIMC tuning rules presented in ref 14) has been employed. We now address the problem of estimating the disturbance transfer function G̃ (s) (see eq 8) for which a new methodology has to be applied. Consider again the closed-loop control scheme of Figure 1 where the PID controller C has been already (possibly roughly) tuned. The measurable step disturbance d of amplitude Ad is then assumed to affect the process output at the time t0 (note that if the disturbance signal is not a step, its dynamics can be included in any case in G(s)). By considering the final value of the integral of the control error, it can be easily be proven that lim

sTq i (s )

lim

(18)

T = T0 − L

−

Thus, by evaluating the final value of the integral of w when d(t) is a step signal of amplitude Ad (namely, d(s) = Ad/s), we can write

Ti ∞

μkK pc(s)e−sL0

H(s) = k ff

(21) 2766

(1 + sTz)e−ϕs 1 + sTp

(26)



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the already implemented compensator. Then, by assuming again that the we have already estimated the process parameters, we have μ = −kk ff −

kK p TA i d

∫0

∞

e(t ) dt

(27)

and τ0 = T0 −

kk ff 1 (Tp + ϕ − Tz) + μ μAd

∫0

∞

w(t ) dt

(28)

where w(t ) = ku(t ) + (kk ff + μ)Ad − y(t )

(29)

It is worth noting at this point that the overall estimation procedure is computationally very simple (it basically requires a few summations) and can be therefore applied on a large scale in a plant with a standard Distributed Control System without a significant increment of the implementation costs.

Figure 3. Results of example 1. Dotted line, initial tuning; dashed line, retuned PID plus feedforward compensator (designed with process and disturbance transfer function estimation by using the PRBS testing); solid line, retuned PID plus feedforward compensator (proposed automatic tuning method).

4. SIMULATION RESULTS In all the following simulation examples we set the amplitude of the set-point and load disturbance step signals as As = Ad = 0.5. After having evaluated a set-point step response and a disturbance step response with an initial tuning of the PID controller and of the feedforward block, the parameters of the process and disturbance transfer function have been estimated. Then, the method developed in ref 9 has been employed to retune the PID controller, and the method presented in Section 3 has been used to design the feedforward compensator. Another experiment has been subsequently performed to verify the effectiveness of the methodology. 4.1. Example 1. As a first example the following systems are considered: P(s) =

e − 2s (1 + 10s)(1 + s)2

G (s ) =

e − 4s 1 + 20s

set-point response and from IAE = 9.966 to IAE = 0.272 for the disturbance rejection task. Note that by employing the method proposed in ref 13 as it is θ > L, we obtain the same results (see eq 15) where τ = Tp). To verify the effectiveness of the proposed automatic tuning method, in particular with respect to the estimation part, a stateof-the-art identification procedure has been applied to the process and disturbance transfer function (separately) and then the feedforward block has been designed accordingly. More specifically, a pseudorandom-binary-signal (PRBS) has been applied in open loop (and in the absence of noise) to the process transfer function and to the disturbance transfer function, and in each case the Matlab System Identification Toolbox function procest has then been applied to estimate the system parameters.18 It results in k = 1.025, T = 12.075, L = 3.47, μ = 1, τ = 20 and θ = 4. The load disturbance response obtained with the feedforward compensator designed by using these estimated transfer functions is plotted (as a dashed line) in Figure 3. The resulting integrated absolute error is IAE = 0.231, which is a very similar value to those obtained previously, thus confirming the effectiveness of the proposed method which requires only routine operation data, that is, closed-loop step responses. Note that the same (retuned) PID controller has been obtained in the comparative example and therefore the set-point step response is the same. 4.2. Example 2. The following models are now considered:

(30)

Initially, the PID parameters are Kp = 0.5, Ti = 10, and Td = 1 and no feedforward action is employed, namely, H(s) = 0. The corresponding step responses are shown in Figure 3 as dotted line (the set-point and disturbance step responses are shown in two different plots for the sake of clarity and both the process and control variable are shown). By applying the proposed estimation procedure, based on the set-point step response, the process parameters have been calculated as (see eq (17), (18), and (19)) k = 1, T0 = 14, L = 3.07, and T = 10.93 while, based on the load disturbance response, the disturbance transfer function parameters have been calculated as (see (21), (24) and (25)) μ = 1, τ0 = 23.99, θ = 4.43, and τ = 19.56. Then the retuning algorithm gives Kp = 1.125, Ti = 6.907, and Td = 4.022 while the feedforward compensator transfer function results in H (s ) = −

1 + 10.93s −1.36s e 1 + 19.56s

(31)

P(s) =

e − 2s (1 + s)3

G (s ) =

e − 4s 1 + 0.5s

(32)

the initial PID parameters are Kp = 0.8, Ti = 3, and Td = 1 and H(s) = 0. Correspondingly, the parameters obtained by using the estimation procedure are k = 1, T0 = 5.002, L = 2.26, T = 2.742, μ = 1, τ0 = 4.504, θ = 4.03, and τ = 0.474. Then the retuning algorithm gives Kp = 0.340, Ti = 1.539, and Td = 1.203 while the feedforward compensator transfer function results in

Figure 3 shows the corresponding results where it is evident that the control performance improves after the retuning of the PID controller and with the use of the feedforward compensator block. Indeed, the integrated absolute error improves from IAE = 9.998 (obtained with the initial tuning) to IAE = 3.389 for the 2767


Industrial & Engineering Chemistry Research H (s ) = −

1 + 2.742s −1.77s e 1 + 0.474s

Article

(33)

Figure 4 shows the performance improvement that results from the retuning of the PID controller and from the use of the

Figure 5. Results of example 3. Dotted line, initial tuning; solid line, retuned PID plus feedforward compensator; dashed line, retuned PID plus feedforward block retuned as in ref 13.

H(s) = −0.83 Figure 4. Results of example 2. Dotted line, initial tuning; solid line, retuned PID plus feedforward compensator.

e − 4s (1 + 10s)(1 + s)2

G (s ) =

e −s 1 + 5s

(36)

The corresponding IAE for the disturbance rejection is 2.004. Obviously, for set-point following task the result is exactly the same. 4.4. Example 4. As a fourth example, we consider the case where an initial (only proportional) feedforward compensator is already in place when the estimation procedure is performed. The process and disturbance transfer functions are, respectively,

feedforward compensator block. Indeed, the integrated absolute errors are IAE = 2.575 and IAE = 2.509 with the initial tuning and IAE = 2.507 and IAE = 0.998 after the application of the proposed method for the set-point and disturbance response, respectively. As in Example 1, the method proposed in ref 13 gives the same result. 4.3. Example 3. As a third example, another third-order process is considered, but in this case the disturbance dead time is smaller than the process dead time: P(s) =

1 + 11.0s 1 + 2.51s

P(s) =

e − 2s (1 + 4s)2

G (s ) =

e −s (1 + 2s)2

(37)

The initial PID parameters are Kp = 0.5, Ti = 8, and Td = 0 and a feedforward block H(s) = −1 is employed. By evaluating the setpoint response (where IAE = 8.0), the following process parameters are obtained: k = 1, T0 = 10, L = 3.23, and T = 6.67. Then, by evaluating the disturbance step response (where IAE = 3.23), the parameters obtained are (see eq 27 and 28) μ = 1, τ0 = 5.0, θ = 1.46, and τ = 3.54. Then, the retuning algorithm gives Kp = 3.54, Ti = 5.17, and Td = 1.60, while the feedforward compensator transfer function results in (note that α = 0)

(34)

The initial PID parameters are again Kp = 0.5, Ti = 10 and Td = 1. No feedforward compensator is initially used. After the evaluation of the set-point and disturbance step responses, the parameters obtained are k = 1, T0 = 16, L = 5.07, T = 10.93, μ = 1, τ0 = 6.0, θ = 1.13, and τ = 4.87. Then the retuning algorithm gives Kp = 0.653, Ti = 6.630, and Td = 4.360, while the compensator transfer function results in 1 + 10.97s H (s ) = (35) 1 + 4.82s

H (s ) = −

1 + 6.67s 1 + 3.54s

(38)

Results are plotted in Figure 6 where the improvement obtained by using a lead-lag feedforward compensator instead of a simple proportional one can be appreciated (note that, after the retuning, it is IAE = 3.66 for the set-point step response and IAE = 1.25 for the disturbance step response).

Note that no delay has been included in the feedworward action (namely, α = 0) and the ideal compensation cannot be achieved. Figure 5 shows the performance improvement that results from the retuning and the feedforward compensator block. The values of the integrated absolute errors are IAE = 9.991 and IAE = 9.979 with the initial tuning and IAE = 5.539 and IAE = 3.158 after the application of the proposed method for the set-point and disturbance response, respectively. The results obtained from the application of the method described in ref 13 are also shown. The feedforward compensator results to be

5. EXPERIMENTAL RESULTS Experimental results have been obtained by employing laboratory scale equipment in which a level control task can be implemented. In particular, two cascade tanks (which are part of the quadruple tanks apparatus made by Quanser) have been employed (see Figure 7). The level of the lower tank is controlled 2768



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Figure 8. Sketch of the experimental setup.

Figure 6. Results of example 4. Dotted line, initial tuning; solid line, retuned PID plus feedforward compensator.

Figure 9. Experimental results with the initial tuning and without feedforward: left part, set-point following task; right part, load disturbance rejection task.

Figure 7. The Quanser apparatus used in the experiments. Only the two tanks in the left part have been employed.

by manipulating the inflow of the upper tank. As a disturbance, an additional inflow to the lower tank has been used (the step-like disturbance has been applied when the set-point step response has attained the steady-state value for a better evaluation of the results), as it is shown in Figure 8. For the sake of simplicity, all the variables are expressed in volts. Initially, the PID controller has been tuned with Kp = 0.1, Ti = 20, and Td = 1. A unit set-point step response (As = 1 V) and a step load disturbance of amplitude Ad = 0.5 V (starting respectively from the steady-state values of y(t) = 2.5 V and y(t) = 3.5 V) have been evaluated (see Figure 9), obtaining IAE = 52.90 and IAE = 57.74, respectively, as values of the integrated absolute error. The parameter estimation procedure has given k = 11.3, T0 = 74.55, L = 6.01 (as a consequence, T = 68.54), μ = 5.17, τ0 =

Figure 10. Experimental results with the retuned PID controller and with feedforward: Left part, set-point following task; right part, load disturbance rejection task.

74.45, θ = 4.01 (as a consequence, τ = 70.44). It appears that the dynamics of the upper tank is basically modeled as a dead time of 2 s. The retuning algorithm proposed in ref 9 has then been applied, resulting in Kp = 0.1, Ti = 13.56, and Td = 54.98 (note that the high value of the derivative time constant with respect to 2769



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Learned and New Approaches; Vilanova, R. , Visioli, A., Eds.; Springer: London, UK, 2012. (8) Veronesi, M.; Visioli, A. Automatic feedforward tuning for PID control loops. Proc. Eur. Control Conf. 2013, 3919−3924. (9) Veronesi, M.; Visioli, A. Performance assessment and retuning of PID controllers. Ind. Eng. Chem. Res. 2009, 48, 2616−2623. (10) Veronesi, M.; Visioli, A. An industrial application of a performance assessment and retuning technique for PI controllers. Int. Soc. Autom. Trans. 2010, 49, 244−248. (11) Veronesi, M.; Visioli, A. Performance assessment and retuning of PID controllers for integral processes. J. Process Control 2010, 20, 261− 269. (12) Veronesi, M.; Visioli, A. A simultaneous closed-loop automatic tuning method for cascade controllers. IET Control Theory Appl. 2011, 5, 263−270. (13) Guzmán, J. L.; Hägglund, T. Simple tuning rules for feedforward compensators. J. Process Control 2011, 21, 92−102. (14) Skogestad, S. Simple analytic rules for model reduction and PID controller tuning. J. Process Control 2003, 13, 291−309. (15) Ang, K. H.; Chong, G.; Li, Y. PID control systems analysis, design, and technology. IEEE Trans. Control Systems Technol. 2005, 13, 559− 576. (16) Åström, K. J.; Hägglund, T. Advanced PID Control; ISA Press: Research Triangle Park, USA, 2006. (17) Åström, K. J.; Hägglund, T.; Hang, C. C.; Ho, W. K. Automatic tuning and adaptation for PID controllersA survey. Control Eng. Practice 1993, 1, 699−714. (18) Ljung, L. System Identification Toolbox User’s Guide; The Mathworks: Natick, USA, 2013.

the integral time constant is not an issue as the PID controller is in series form). The feedforward block results: H(s) = −0.458

1 + 68.55s 1 + 70.45s

(39)

It has to be noted that as the pole and the zero are very similar, a simple gain could be employed instead of the lead-lag block without impairing the performance significantly. Further, the feedforward block determined by using the technique presented in ref 13 gives virtually the same result, as the only difference is the gain of the lead-lag block which results to be kff = 0.436. The step responses (both for the set-point and the load disturbance) obtained with the automatically redesigned control system is shown in Figure 10, where it is IAE = 14.61 for the set-point step response and IAE = 4.11 for the load disturbance step response. By considering the different scales of the plots, the improvement of the performance is evident (actually, the PID controller has been initially tuned very badly).

6. CONCLUSIONS We have proposed a methodology for the closed loop identification of the (FOPDT) transfer function through which a process disturbance affects the controlled variable of a PID control system. The main parameters of the transfer function are automatically determined by integral computations which are robust with respect to the measurement noise. It has to be remarked that the procedure can be applied to data available during routine operations without stopping the running control loop and that the results do not depend on the employed PID parameters. As a result, a classical feedforward compensator can be automatically designed and included in the control strategy. The effectiveness of the overall methodology has been demonstrated by simulation and experimental results. The methodology can be implemented by means of standard control instrumentation.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +39-030-3715460. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the reviewers for the useful suggestions on how to improve the paper. REFERENCES

(1) O’Dwyer, A. Handbook of PI and PID Tuning Rules; Imperial College Press: London, UK, 2006. (2) Visioli, A. Practical PID Control; Springer: London, UK, 2006. (3) Visioli, A. A new design for a PID plus feedforward controller. J. Process Control 2004, 14, 455−461. (4) Petersson, M.; Arzen, K.-E.; Hägglund, T. A comparison of two feedforward control structure assessment methods. Int. J. Adapt. Control Signal Process. 2003, 17, 609−624. (5) Vilanova, R.; Arrieta, O.; Ponsa, P. IMC based feedforward controller framework for disturbance attenuation on uncertain systems. Int. Soc. Autom. Trans. 2009, 48, 439−448. (6) Rodriguez, C.; Guzmán, J. L.; Berenguel, M.; Hägglund, T. Generalized feedforward tuning rules for non-realizable delay inversion. J. Process Control 2013, 23, 1241−1250. (7) Guzmán, J. L.; Hägglund, T.; Visioli, A. Feedforward compensation for PID control loops. In PID Control in the Third Millennium: Lessons 2770


Automatic Tuning of Feedforward Controllers for Disturbance Rejection

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