FFT-based monitoring procedure for intelligent control

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Pergamon

Chemical En~tineeriny Science, Vol. 52, No. 16, pp. 2823 2828, 1997

PII:

S0009-2509(97)00090-0

i~ 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0009-2509/97 $17.00 + 0 . 0 0

FFT-based monitoring procedure for intelligent control (Received 27 July 1996; in revised form 26 J a n u a r y 1997, accepted 14 M a r c h 1997)

INTRODUCTION In the chemical process industries, many control schemes are made up of PID/PI controllers. The main reason for this is their relatively simple structure which is easy to understand and to implement. Although they are widely used in process industries, many PI/PID control loops perform poorly. This is because the tuning is based on the process model which is identified near one particular operating condition. Since most chemical processes are usually nonlinear and are often operated at different operating conditions, the mismatch between the real process and the process model often arises. As a result, the stability and the performance of the control system will change with time after the commission if little attention is paid to monitor whether the controller is performing according to the design specification. This is the reason why the monitoring scheme is a vital component in many control systems. The goal of monitoring is to provide the information that can be used to decide whether there is sufficient incentive to redesign the controller. Monitoring plays an important role in the intelligent control which has attracted a great deal of interest in the last decade (Arzen, 1989; Astrrm et al., 1986; Doraiswami and Jiang, 1989; Ortega et al., 1992). However, current monitoring measures in expert control are obtained mainly in time domain, and the deterioration of the control system cannot be detected until a significant deviation in the controlled variable is observed. Therefore, an efficient monitoring method is necessary to maintain the tight product quality specification. As mentioned above, the dynamics of many chemical processes change with time and therefore the controller design has to take this plant/model mismatch into account, i.e. robust controller design. The concept of maximum closed-loop log modulus (Luyben, 1990), L ...... provides a good measure of robustness for single-input-single-output (SISO) systems. The control system with a smaller Lc.m,x means it can tolerate a larger process/model mismatch, but with a more sluggish response. Since a reasonable control system should achieve an acceptable performance as well as the stability in the presence of modeling errors and other uncertainties, a certain trade-off must be made between performance and robustness. Based on this, a + 2 dB maximum closed-loop log modulus is recommended as design criterion for SISO systems (Luyben, 1990). A large number of tests show that this criterion yields a reasonable compromise between robustness and performance. The purpose of the proposed method is to allow engineers to assess the current status of the existing control system by identifying L ..... on-line, to monitor how Lc.ma x changes with time,

and to assist in deciding whether a redesign of the controller is necessary. Therefore, if L ..... is measured on-line once in a while, we can have the up-to-date robustness measure for the control system. More importantly, the proposed monitoring method is a frequency domain method, which can detect the robustness of a control system before a disturbance comes into the system, whereas the time domain monitoring method can not assess the robustness of a control system until a significant deviation in the controlled variable is observed. In order to evaluate L ..... on line, Chiang and Yu (1993) use the relay feedback technique to calculate Lc,ma~ with two to three relay experiments for SISO systems. Ju and Chiu (1996) further extend Chiang and Yu's method to two-input-two-output (TITO) control systems. But, for some control systems, it is time consuming to apply their methods in practice due to the tedious procedure involved. Therefore, a more efficient and accurate monitoring method is required from practical point of view. The recently developed Fast Fourier Transform (FFT)-based frequency response identification technique (Wang et al., 1996) is capable of achieving this objective. The theory behind FFT technique reveals that as many accurate frequency response points as desired can be obtained from one relay experiment. Because of the simplicity and accuracy of this method, the proposed FFTbased monitoring procedure has a great potential in industrial application. RELAY FEEDBACK SYSTEMS AND Lc,,,~x Astr~m and H~igglund's relay tuning (1988) is a simple method for exciting a dynamic system and estimating useful process characteristics which can be used as the basis for the Ziegler-Nichols tuning method (Ziegler and Nichols, 1942). Figure 1 illustrates the relay feedback system in which an ideal relay (N) is inserted in the feedback loop. The key idea is that many systems will exhibit stable limit cycles when subjected to relay feedback. Figure 2 shows typical responses in process input (u) and process output (y) in a relay feedback experiment. According to Luyben (1990), the maximum closed-loop log modulus, L~.,,ax, is defined as follows for SISO control systems, G(jo~)K(j~o) L ..... = 201oglo max 1 + G(j~o)K(j~o) ]

(1)

where G and K denote the process and the controller transfer function, respectively. Lc.ma~ provides a good robustness measure for control systems. In fact, L ..... relates to the

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N set

U

.r "1

d rl

G

Y

Fig. 1. Astr6m-H~igglund relay feedback system.

DISCRETE FOURIER TRANSFORM AND FAST FOURIER TRANSFORM

Y

time

Suppose t h a t f ( t ) is a continuous function and is sampled at the time interval of T. The sampled values f,~ -~f(mT), m = 0, 1, 2, ..., represent the magnitude of the continuous signal at each sampling instant. If the functionf(t) is nonzero only in a finite interval of time, that is,f (roT) = 0 for m t> L, we have the definition of the Discrete Fourier Transform (DFT) of the L points fro (Brigham, 1974; William et al., 1989), which is denoted by Fi L-1

L-I

Fi = F(mi) =- T ~ f,.e -2~jmi/L = T ~ f,.e -j'°'"r, m=0

m=O

i=O,l ..... L-

1,

(2)

where the frequency ml is defined as

U

2hi

mi=~,

"If I

i=O, 1.... , L - 1 .

The formula for the Discrete Inverse Fourier Transform, which recovers the set of fin's exactly from the F~'s is

time

Fig. 2. Relay feedback experiment.

H ~ norm of the complementary sensitivity function, n(j~o) = G(jo~)K(jog)/[1 + G(j~o)K(jc~)], for SISO control systems. In H ~ control theories, this value is a good indicator for the design constraint imposed by the robustness consideration (Zames, 1981; Morari and Zafiriou, 1989). Since a tighter control means a less robust of the system and vice versa, a trade-off must be made between the performance and robustness. The measured L~. . . . from the proposed method can be compared with + 2 dB, which is used as a design criterion for SISO systems by Luyben (1990). If the measured L . . . . . is larger than + 2 dB, the control system will be less robust but with a tighter control. Thus, the gain of the controller needs to be decreased or the reset time of the controller needs to be increased. On the contrary, if the measured Lc.maxis less than + 2 dB, the control system will be more robust but with a more sluggish setpoint and load response. Thus, we need to increase the gain of the controller or decrease the reset time of the controller. In fact, this kind of tuning scheme is termed as BLT method (Luyben, 1986). It is noted that the BLT method is easy to use by control engineers and leads to settings that compare very favorably with the empirical settings found by exhaustive and expensive trial-and-error tuning methods. Thus, the proposed method can be used in conjunction with the BLT method in on-line tuning the controller. In conclusion, the measured Lc.max from the proposed method can indicate whether the process shifts away from the original design condition. If this is indeed the case, attention should be focused on the redesign of the controller.

1 L~- I Fi e2'~jmi/L= ~ T L -~1 Fie j~''mT, fra =---~ i=0

m = 0, 1. . . . . L -

i=0

1.

(3)

It is obvious that, the more sampling points there are, the longer computing time will be required. The efficient tool to calculate the D F T is the Fast Fourier Transform (FFT). There are different versions of FFT implementations available. The FFT routine used in our program is based on one originally written by N. Brenner of Lincoln Laboratory (William et al., 1989), from which the Inverse Fourier Transform can also be obtained with slight modification.

FFT-BASED MONITORING PROCEDURE Figure 3 is the configuration of the monitoring scheme studied by Chiang and Yu (1993). The objective of this monitoring scheme is on-line identification of L . . . . . . For the purpose of monitoring, a relay N is placed between the process G and the controller K. In the monitoring mode, the controller output goes directly into the relay and the input of the process is the output of the relay. In the control mode, there is no relay and the controller output goes into process directly. Instead of searching over the whole frequency range to find L . . . . . . which requires both the information of process and controller, Chiang and Yu (1993) and Ju and Chiu (1996) use several relay experiments to obtain L . . . . . on-line for SISO systems and TITO systems, respectively. However, their methods involve several relay feedback experiments which can be tedious in industrial application. In this paper, Fourier Transform technique is employed such that L . . . . . can be obtained directly from only one ideal relay feedback experiment. The principle behind this is analyzed as follows. From the definition of D F T given in eq. (2), it is obvious that the signal to be transformed should be nonzero only in a finite interval of time. That is to say, the signal should be absolutely integrable. This means that the input v(t) and output u(t) shown in Fig. 3 can not be

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N set

[

K

m

+

© e

Fig. 3. Relay feedback monitoring system: m standing for monitoring mode, c standing for control mode.

transformed to frequency response by directly using FFT. This problem can be overcome by introducing a decay exponential e ,t to form (Wang et al., 1996) -- f(t)

=

-- v(t)e

Therefore, the information of K(jtn)G(jto) can be obtained from the following steps: (k'-g),, = F F T - I ( K ( j c o + c0G(jo + ~)),

(12)

(4)

-~',

where FFT - ~ denotes the Inverse Fourier Transform. From eq. (5), the next relation is obvious:

and ti(t) = u(t)e -~'.

(5)

Suppose that v ( m T ) and u ( m T ) , m = 0, 1. . . . . L - 1, are samples of v(t) and u(t), respectively. L is chosen such that both v((L - 1)T) and u((L - 1)T) have reached the stationary state, and the decay coefficient ~t is chosen such that both f((L - 1)T) and fi((L - 1)T) have decayed to zero. According to Wang et al. (1996), ~ can be chosen such that In ), ~> - - tf

(6)

where y is the specified threshold and usually takes the value of 10-5-10 - :0, and t I is the experimental time span. Applying the Fourier Transform to - ~(t) and t~(t) yields - 17(jo) =

- f(t)e-JO, tdt =

_ v( t)e- ,, e - i,~ dt

= - V(jo) + c0,

(7)

and t2(j~)) =

;

a(t)e-J~"dt =

;o

u(t)e-"e

IT(j~) /.7(jo)'

= -- T ~

v ( m T ) e-j'o'"r = -- FFT(g(mT))

(10)

m=O

and /.7(jcol) = T ~ ti(mT)e -~'~'mr m=0

a ( m T ) e -j'°''~r = FFT(ti(mT)). rn=O

(15)

EXAMPLES Example 1 The control system under consideration is adopted from Chiang and Yu (1993), where G1 (s) =

(7200s + 1X2s + 1)

and 1 KI (s) = 10.54 + s.~l~'.-^--"

~ ( m T ) e -i~'"T

L-1

r

(14)

37.3e- 102

m=0

-- T Z

ego ~< (Z - 1)¢o0,

(9)

where 17(j~o) and /.7(jw) can be computed at discrete frequencies (ol) with FFT technique, which are -- 17(jtoi)

K(j¢o,)G(joJ~) = FFT((kg),,).

Thus, the procedure of identifying L . . . . . can be summarized as follows: Step 1: Form - f(t) and ti(t) by using eqs (4) and (5); Step 2: Apply the FFT technique to - f ( m T ) and a ( m T ) to obtain discrete frequency response points via eqs (t0) and (11); Step 3: Use eq. (9) to calculate the shifted frequency response of K(jt, + :t)G(jto + ~t); Step 4: Calculate frequency response of K G from eqs. (12)-(14); Step 5: Obtain L ..... from eq. (1). Thus, based on one ideal relay experiment with the oscillation frequency at o)c, L . . . . . can be identified. The number of frequency points, Z, which will be used in the calculation of L . . . . . can he obtained by

where ¢Oo is the frequency increment desired by users.

(8)

V ( j ~ + ~) U(jco + a)

(13)

Finally, the frequency response of K G can be obtained from the values of (kg),.'s via FFT technique.

J'°'dt = g ( j o + ~)

It is clear from the monitoring system in Fig. 3 that the relation K(s)G(s) = -- V(s)/U(s) holds. As a result, the shifted frequency response K ( j ~ ) + ~)G(jco + ~) can be obtained as K(jo9 + ~)G(j~o + a) =

(kg)m = (k-g)me""~r.

(ll)

To employ FFT method to monitor this control system, an ideal relay experiment is performed and the data of v(t) and u(t) are logged as shown in Fig. 4. The resulting L . . . . . (shown in Fig. 5) is 10.481 dB which is almost identical to 10.440 dB obtained by Chiang and Yu (1993) by employing three relay experiments. The identified frequency responses of G K obtained from FFT technique and those from Chiang and Yu's method are shown in Fig. 6. From the above-mentioned comparisons, it is clear that one can use only one relay experiment to on-line identify L . . . . . instead of several relay experiments required by Chiang and Yu's

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0.8 1

0.6

0.8

0.4

0.6 0.4

0.2

0'2 0

>

0

0;I I

-0.2

0.4

-0.4

O6 0

-0.6 -0.E

200 ' time

400

0

2~o

400

time

Fig. 4. Ideal relay experiment of Example 1.

12 10

4

0

0.06

I

I

I

I

0.07

0.08

0.09

0.1

0.11

Frequency Fig. 5. Closed-loop log modulus versus frequency resulting from FFT technique for Example 1.

method. Furthermore, the monitoring time can be saved greatly, which is very important from intelligent control point of view. The sensitivity of the proposed method to the measurement noise is also studied. If the white noise is inserted between the controller K and relay element N, and the ratio of the noise to signal is set to be 5%, the identified Lc, max is equal to 11.039 dB, which is comparable to the result obtained in the absence of noise. It means that the proposed FFT-based identification method can handle the process noise to some extent. Example 2 The on-line search method for Zc,ma x proposed by Chiang and Yu (1993) is confined to the thirdquadrant. Therefore, if the L . . . . . is located outside the third quadrant, e.g., in the second quadrant, their method will fail. Certainly, this problem can be overcome by inserting some artificial elements,

e.g., a derivative term "s", a fictitious transportation lag and so on, such that the search space for the phase angle falls in the third quadrant instead of in the second quadrant. However, this scheme is not desirable because it will make the control system rather complicated. One important feature of the proposed FFT-based identification method is that the search space of Lc,m=, is not limited in the third quadrant. As many as desired frequency points can be obtained to facilitate the calculation of Lc.~,. To illustrate this feature, we choose a control system to be same as that of Example 1, except the deadtime of the process is changed. 37.3e- 202 G,(s) =

(7200s + 1)(2s + 1)

1 K l ( s ) = 10.54 + 3.79s"

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2827

0.05 0 -0.05 -0.1 2" -0.15 E -o.2 -0.2! -0.: -0.3.~ .0, t

-1.1

I

-1

I

-0;9

-0.8 Real (GK)

-017

-016

-0.5

Fig. 6. Part of the Nyquist curve of GK generated by Chiang and Yu's method (solid line) and FFT technique (circle). 21

20 19 El

~18 4 17

16

15 0.045

' 0.055

'

0.05

0.06

Frequency Fig. 7. Closed-loop log modulus versus frequency resulting from FFT technique for Example 2.

From eq. (1), it can be computed that the Lc.ma. of this control system is 20.388 dB, which occurs at the frequency located in the second quadrant. However, FFT-based monitoring method is still valid for this case. Figure 7 demonstrates that the resulting Lc.... from one relay feedback experiment is 20.234 dB, which differs from the actual value found in frequency domain by less than 1%. Therefore, from practical point of view, the proposed method is more versatile than Chiang and Yu's method. CONCLUSION Performance monitoring plays an important role in intelligent control system. A FFT-based monitoring method is proposed. Compared with the previous monitoring methods, it has several advantages. Firstly, it is a frequency domain method, which can evaluate the performance of the controller before the disturbance comes into the system. Secondly, the method is very effective and convenient, and the testing

time can be saved greatly, since only one ideal relay experiment is involved. Lastly, the frequency search range for on-line identification of L ..... is extended, whereas Chiang and Yu's method (1993) is restricted to the third quadrant. Simulation also shows that the proposed method can handle the process noise to some extent. All these features will make this FFT-based monitoring method very attractive in the intelligent control area. JUN JU MIN-SEN CH1U* QING-GUO WANG

Department of Chemical En~lineering National University of Singapore 10 Kent Ridge Crescent Sinoapore 119260 Singapore * Corresponding author.

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Shorter Communications NOTATION

DFT FFT Fi

f.

G K L

Le,max N SISO T TITO tf

u(t) v(t)

Z

Discrete Fourier Transform Fast Fourier Transform Discrete Fourier Transform of the L points f,, sampled value off(t) at t = m T process transfer function controller transfer function number of the sampled points maximum closed-loop log modulus describing function of relay single-input-single-output sampling interval two-input two-output experimental time span process input controller output number of frequency points used in the calculation of L ..... by FFT

Greek letters c~ decay coefficient 7 specified threshold (10-5-10-10) 030 frequency increment desired by users ~oc oscillation frequency measured from ideal relay experiment REFERENCES

Arz~n, K. E. (1989) An architecture for expert system based on feedback control. Automatica 25, 813 827. Astr6m, K. J., Anton, J. J. and Arz6n, K. E. (1986) Expert control. Automatica 22, 277-286. Astr6m, K. J. and Hiigglund, T. (1988) Automatic Tuning of PID Controllers. Instrument Society of America, Research Triangle Park.

Brigham, E. Oran. (1974) The Fast Fourier Transform. Prentice-Hall, Englewood Cliffs, NJ. Chiang, R. C. and Yu, C. C. (1993) Monitoring procedure for intelligent control: on-line identification of maximum closed-loop log modulus. Ind. Engng Chem. Res. 32, 90-99. Doraiswami, R. and Jiang, J. (1989) Performance monitoring in expert control systems. Automatica 25, 799-811. Ju, J. and Chiu, M. S. (1996) On-line performance monitoring of two-inputs-two-outputs multiloop control systems. Comput. Chem. Engng 20, s829-s834. Luyben, W. L. (1986) Simple method for tuning SISO controllers in multivariable system. Ind. Engng Chem. Process Des. Dev. 25, 654-660. Luyben, W. L. (1990) Process Modelling, Simulation, and Control for Chemical Engineers. McGraw-Hill, New York. Morari, M. and Zafiriou, E. (1989) Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ. Ortega, R., Escobar, G. and Garcia, F. (1992) To tune or not to tune?: A monitoring procedure to decide. Automatica 28, 179-184. Wang, Q. G., Hang, C. C. and Zou, B. (1996) A Frequency response approach to autotuning of multivariable PID controllers. Proceedings of 13th World Congress of lFAC, Vol. K, pp. 295 300. San Francisco, U.S.A. William, H. P., Brian, P. F., Saul, A. T. and William, T. V. (1989) Numerical Recipes: The Art of Scientific Computing (Fortran Version). Cambridge University Press, Cambridge. Zames, G. (1981) Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms and approximate inverses. IEEE Trans. Autom. Control AC26, 301-320. Ziegler, J. G. and Nichols, N. B. (1942) Optimum setting for automatic controllers. Trans. ASME 64, 759 768