New frequency-domain design method for PID controllers

New frequency-domain design method for PID controllers

L. Wang, PhD T.J.D. Barnes, MSc W.R. Cluett, PhD

Indexing terms: P I D control, Frequency-domain design

Abstract: This paper presents a new PID controller design method based on process frequency response information. The novel ideas lie in the way that the closed-loop performance is specified via the desired response of the control signal, and in the use of only one (for PI control) or two (for PID control) process frequency response points in the design. Straightforward analytical formulas are given for the PID controller parameters. Simulation studies are given to compare this design method with other design methods found in the literature. The results indicate that the new method provides much smoother responses in both the control signal and process output, which are generally more desirable in the process control setting.

1 introduction

The PID controller continues to be the most common type of controller used in the process industries. However, the tuning of these controllers is still not widely understood and many operate with their original default settings. Despite this, researchers continue to search for relatively simple ways to design these controllers to improve closed-loop performance. However, it is safe to say that not one method in over 50 years has been able to replace the Ziegler-Nichols [l] tuning methods in terms of familiarity and ease of use.

Recent developments in the area of PID controller tuning fall into three categories. The first category consists of model-based controller design methods. With this approach, a structured model of the process (typically a Laplace transfer function) is used directly in a design method such as pole-placement or internal model control (IMC) to derive PID settings, with the user specifying performance in terms of some desired closed-loop time constant. These approaches to PID design carry restrictions on the allowable model structure, although it has been shown that a wide range of types of processes can be accommodated if the PID controller is augmented with a first-order lilter in series. An example of this design approach may be found in Reference 2. The second category consists of design methods where the

Q IEE, 1995 Paper 1859D (C8, C9), first received 27th July 1994 and in revised form 3rd January 1995 The authors are with the Department of Chemical Engineering, Uni- versity olToronto, Toronto, Canada MSS 1A4

I E E Proc.-Control Theory Appl., Vol. 142, No. 4, July 1995

PID controller parameters are found by optimising some integral feedback error performance criterion. This approach can be applied to different transfer function models, and typically a numerical search is used to find the optimal parameters. Reference 3 is a recent example of this approach. The third category includes design methods which make use of process frequency response data. Perhaps motivated by the wide acceptance of the Ziegler-Nichols frequency response method, which requires knowledge of only one point on the process Nyquist curve, ways have been developed to automate the Ziegler-Nichols method [4], to refine the Ziegler- Nichols tuning formulas [SI and to develop improved design methods which require only a slight increase in the amount of process frequency response information [6,7l.

From our point of view, the various approaches each have their advantages. The first two model-based approaches have a more intuitive time-domain performance specification than the traditional frequency- domain design methods. However, the frequency-domain methods require less structural information about the process dynamics. In this paper it is our objective to combine these advantages into one single PID design method. We propose to use a limited number of points on the process Nyquist curve without requiring any structural information about the process dynamics other than knowledge of whether the process is self-regulating. Another feature of this design method is that we will use a time-domain performance specification on the closed- loop behaviour of the control signal rather than on the controlled output or the feedback error. We feel that this choice has advantages when using only a limited amount of process frequency response information in the design because it does not rely on any direct pole-zero cancel- lation. In addition, the behavour of the controller output is an important consideration when assessing overall closed-loop performance in a process control application [8]. Since we plan to make use of only a limited number of points on the Nyquist curve, we will address the ques- tion of which points have the largest impact on the closed-loop time-domain performance, and therefore which should be used in the design. We will also derive

The authors wish to acknowledge financial support from the Government of Canada’s Network of Centres of Excellence Program. We would also like to thank Dr. N. Leonard Segall at Imperial Oil Ltd. in Sarnia, Canada, for his sug- gestions on control signal specifications.

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straightforward analytical solutions for the PID parameters to put our results on a comparable footing with the Ziegler-Nichols tuning formulas in terms of ease of use. Finally, we will present two simulation examples to provide some insight into our method and to compare it with a cross-section of other design methods found in the literature.

2 Control signal specification

One of the most common features of many PID controller designs is that performance is specified in terms of the trajectory of the desired closed-loop process output response to a setpoint change. We propose to specify the closed-loop performance in terms of the desired behaviour of the control signal to a setpoint change.

Consider the feedback system illustrated in Fig. 1, where U and y are the control signal and measured

Fig. 1 1 Block diagram offeedback control system

process output, respectively. r is the setpoint, d is the load disturbance, and C and G denote the controller and plant, respectively. We will assume that the controller C has the structure of a PID controller with the following Laplace transfer function :

c2s2 + CIS + CO C(s) =

S

change required to achieve the new setpoint. It also determines the relative response speed between the open- loop process and the desired closed-loop system. For

* ' O i l l

!/ O5 0.51 , , ~ , , , , , , .-

0 2 4 6 8 10 12 14 16 18 20 normalised time

Fig. 2 Desired control signal trajectoryfor a unit step change in setpoint with r = I and a = 0.S and I , 2 (value of control signal is zero prior to 1 = 0)

instance, when a = 1, the speed of the desired closed-loop system is equal to the open-loop process response. When a < 1, the speed of the desired closed-loop system is slower than the open-loop response; and when a > 1, the speed of the desired closed-loop system is faster than the open-loop response.

From eqn. 4 the transfer function relating the desired control signal to the setpoint is simply

_- U(S) C(S) R(s) - 1 + C(s)G(s)

1 am + 1 G(0) ts + 1 - (5)

and the desired closed-loop transfer function is given by

Y(s) 1 ats + 1 '' R(s) G(0) ts + 1

where K, = clr tI = cl/co, T D = c2/cI, or that of a PI controller G (s)=-=-- G(4

CIS + CO C(s) = -

S (3)

We will now present our PID controller design specifications for the two types of processes most frequently encountered in the process industries.

2.1 Specification for stable processes Here we assume that the process is stable with a known steady-state gain equal to G(0). We introduce a new design parameter a which is related to the desired initial change in the control signal for a given step setpoint change. Then, given some desired time constant t for the control signal response, the trajectory of u(t) for a step setpoint change of magnitude r will be specified as

We can see that by using the control signal specification in eqn. 5, a lead-lag element has been added to the open- loop process transfer function 4 s ) to form the desired closed-loop transfer function in eqn. 6. One way for the user to choose a and t would be to make direct use of eqn. 4. In many cases in the process industries, a choice of a = 1, which corresponds to the specification of a step change in the control signal, is acceptable and does not even require a choice for z (see eqn. 5). Alternatively, if some prior knowledge is available about the dominant time constant of the process T (say from an open-loop step response), we can select at = T. Then, after choosing the relative response speed measure a, we can determine t directly via t = T/a.

2 2 Specification for integrating processes Suppose that the process transfer function G,(s) has the following form:

(4)

Fig. 2 illustrates u(t) with a normalised time constant T = 1 and steady-state gain G(0) = 1 for a = 0.5 and 1, 2. It can be readily verified that, in general, 40) = [ r x a/G(O)] and u(o3) = [r/G(O)]. From this Figure we

in the control signal expressed as a fraction of the total

u(t) = -L [a + (1 - ax1 - e-''')]

1 S (7)

= - c(s)

(8) can see that the parameter a determines the initial change -_ - G(o) (1 + y,s + ' ' .)

266 IEE Proc.-Control Theory Appl., Vol. 142, No. 4 , July 1995

where G(0) and y1 are assumed known. Here we want the controller to have integral action to ensure zero offset for both step setpoint and load disturbances. Therefore the moment expansion of G,,(s) must have the form

C,,(s) = 1 + OS + p2 s2 + . . . (9) for the corresponding desired open-loop transfer function to contain a double integrator. We will select the desired closed-loop transfer function relating the setpoint to the control signal to be of the form

U(S) s (2jr - yl)s + 1 R(s) - G(0) r2s2 + 2jrs + 1

which gives a desired closed-loop transfer function

that satisfies eqn. 9. In this case, the desired response of the control signal to a step change in the setpoint is determined by the time constant 5 and damping factor j . However, in most process control applications it would be desirable to fix = 1 to ensure a nonoscillatory response in the control signal. The initial change in the control signal for a step setpoint change of magnitude r is given by r(2jr - yl)/G(0)r2.

3 PID controller design

3.1 The design algorithm Our objective is to find the PID controller parameters such that the actual closed-loop frequency response is close to the desired closed-loop frequency response Gc,(jw). However, the direct approach to this problem involves a nonliner optimisation procedure. Alternatively, we will examine this problem by working with the open- loop transfer function because in this case the problem becomes linear in the controller parameters. The desired open-loop transfer function is given by

and we define a complex function

= Y R ( ~ + j Y , ( 4 (14) Note that the frequency-domain error between the desired open-loop frequency response God jw) and the actual open-loop frequency response C( jw)G( jw) is zero, if

Y(jw) = cZ(jw)’ + c l j o + c o , for PID (15)

Y(jw) = cljw + co, for PI (16)

or

Hence, to guarantee a small frequency-domain error, the structure of Y(jw) for a PID controller needs to satisfy for omin d w d wwx and some (either all positive or all negative) constants PI, p2 and p3

Y ( j 4 = 82(jo)2 + B1jw + Bo (17)

yR(w) 8 0 - 82 O2 (18)

or in terms of its real and imaginary parts

IEE Proc.-Control Theory Appl., Vol. 142, No. 4, July 1995

and

Y , ( 4 = 810 (19) Eqqs. 18 and 19 indicate that the graph of the real part of Y( jw) versus w2 and the graph of the imaginary part of Y(iw) against w should both behave like straight lines. For a PI controller, the real part of Y(jw) should be reduced to a horizontal line. Based on these characteristics of Y(jw), we can solve for the controller parameters c2, c1 and co by directly fitting straight lines to the real and imaginary parts of Y(jw). Hence, given the process frequency response at just two frequencies G(jol) and G(jw,) with w1 < w2, we can compute the PID controller parameters from the real and imaginary parts of Y( jw) as follows :

For a PI controller, c1 is given by eqn. 21 and

= yR(wl) (23) Using this algorithm, the error between the actual and desired closed-loop frequency reponses is zero at w = 0, w1 and w2 (real part only) for a PID controller. If eqn. 18 and 19 are satisfied, then the errors at other frequencies are guaranteed to be small.

3.2 Choice of frequency points It is known that an arbitrary performance level can be achieved for processes with first-order or second-order transfer functions under PID control [6], where the structure of Y(jw) satisfies eqns. 18 and 19 at all frequencies 191. For more complicated processes, Y(jw) will usually satisfy eqns. 18 and 19 at low frequencies but violate them at high frequencies. The well-known Ziegler-Nichols method is based on the cross-over frequency of the process. The frequencies used in References 6 and 7 in dominant pole design are tuning parameters for closed-loop performance. In this paper our objective is to identify the important frequencies for use in our PID design method by examining the relationship between the desired closed-loop frequency response G,,(jw) and its corresponding unit step response y,(t).

We will assume that the unit step response y,(t) is approximately equal to unity for t > T,, where T, is the desired closed-loop settling time, and that the closed-loop system is sampled with an interval At . It can be shown that there is a unique relationship between the sampled step response yc(ti) ( i = 0, 1, . . . , N - 1 and T, = N A t ) and its discrete frequency response Gf,(eiw) given in Refer- ence [lo].

We have also shown that for small At and therefore large N, Gcl(jwk) FZ Gtl(ej2rk/N) where wk = 2nk/T,. Then we have

Eqn. 25 indicates that the closed-loop frequency response G,,(jw) evaluated at the set of frequencies, w = 0, 2n//T,,

267

. . . ., n/At, determines the step response of the closed-loop system. If Iimmdm I Gcl(jm)l -0, then this set is reduced to a smaller set of lower frequencies.

To choose the two frequencies for our PID controller design, we will examine the effects of the weighting function on the desired closed-loop frequency response in eqn. 25. Let

S,(k, i) = [l-cos($) 2N[1 - cos (2xk /N) ]

+ 2 sin ($) sin ( y + $)] (27)

and an imaginary part

S,(k, i) = [sin (g) 2 N [ 1 - COS (2xk /N) ]

- 2 sin ($) cos (F + $)] (28)

It can be verified that

Hence, S,(k, i ) and S,(k, i) act as weighting functions on the real and imaginary parts of the desired closed-loop frequency response. Fig. 3 shows the behaviour of the

/

v -021 , , I I 1 , . i

0 20 40 60 80 100 120 140 160 180 xx) sampling instants

Fig. 3 Real part ofweightingfunction S(k, i)for N = 200

weighting function S,(k, i) for N = 200 and k = 0, 1, 2, 7 . For k = 0, it can be seen that this weighting function starts at l/N for i = 0 and inreases in a linear fashion until it reaches a value of 1.0 at i = 199. Fig. 4 shows the behaviour of S,(k, i) for N = 200 and k = 1, 2, 7. For k = 0, S,(k, i) is equal to zero for all i. For k > 0, both S,(k, i) and S,(k, i) have their largest magnitudes when k = 1 and hence GC,(j2n/T,) contributes most to the shape of the step response y,(tJ beyond G,,(O). The next most significant term is when k = 2. The weighting functions for k = 7 are presented to indicate that as k increases, the contributions from the corresponding higher frequencies to the step response decrease. Therefore to attempt to

268

achieve the desired closed-loop performance, the first frequency that should be included in the PID design is w l = 2x /T , , and the second is wt = 4n/T, , with G(0) assumed to be known.

I

01

g o 3 P 5 @

0 5 02 - I

B $ 0 1 c_ E E

0 0 20 40 60 80 100 120 140 160 180 200

sampling instants

Fig. 4 Imaginary part of weighting function S(k, i ) for N = 200

3.3 Assessment of achievable performance In general use of a PI or PID controller will constrain the achievable closed-loop performance [SI. The proposed method has the advantage that these constraints can be checked to some degree by examining the properties of Y(jw). Here, we will assume that the process gain G(0) is positive.

3.3.1 Assessment for PI controllers: Let Y(jo) be expressed in polar form, i.e.

Y(jo) = Ym(w)i-jQ(") (30) To ensure that both the proportional gain and integral time constant of a PI controller are positive, the following condition must be satisfied:

(31) 0 < Q(mi) < ( 4 2 ) for the wI used in the PI controller design. In practice, it is desirable to have the sign of the controller gain match that of the process gain. With respect to the integral time constant, Astrom [6] pointed out that a necessary condition for closed-loop stability is that this controller parameter should be positive. If the user finds that a performance level has been specified such that the condition 31 is not satisfied, the performance should be detuned until it is satisfied.

3.3.2 Assessment of PID controllers: To guarantee a positive proportional gain, the following condition must be satisfied:

0 < Q(wi) < (32)

yR(wl) > yR(oZ) (33)

yR(OZ)w: ' yR(wl)O: (34)

For a positive derivative time constant we require

and for a positive integral time constant

where o1 and w2 are the two frequencies used in the PID design.

3.4 Summary of information required for design

3.4.1 Process-related information: Is the process stable or integrating? If the process is stable, an estimate of the

IEE Proc.-Control Theory Appl., Vol. I42, No. I , July 1995

process gain G(0) is required. If the process is integrating, an estimate of G(0) and y1 in eqn. 8 is required. For example, if the process is described by an integrator-plus- delay transfer function, y1 is equal to an estimate of the delay. An estimate is required of the process frequency response at o1 = 2x/T, (for PI design) and at o2 = 4n/T, (for PID design).

3.4.2 Performance-related information: If the process is stable, a choice of a and t is required. The reader is referred back to the discussion below eqn. 6. If the process is integrating, with [ set equal to unity, a choice for T is required. An estimate of the desired closed-loop settling time T, is required to determine the frequencies used in the design.

4 Simulation results

To illustrate our design and compare it with existing design methods, we will use two first-order plus delay models, with the first example having a deadtime-to-time- constant ratio of less than one (0.5), and the second example having a ratio much greater than one (5.0). Both types of processes are frequently encountered in the process industries, and therefore represent a reasonable basis for comparison.

4.1 Example 1 Consider the problem of PID controller design for the following plant transfer function model:

e ~ 5s

G(s) = - 1os+ 1 (35)

We have applied our design method using three. different values of a (0.5, 1.0 and 1.5) to provide insight into the role of this key performance-related parameter. In all cases the parameter T has been selected according to t = T/a where T = 10 for this example. The closed-loop settling times T, have been estimated directly from the open-loop settling time of 55 and the choice of a according to T, = 55/a. The PID controller parameters for these three cases are summarised in Table 1. We have simu-

Table 1 : PID parameters for Example 1

a = 0 . 5 a = l 01-1.5 IMC 2-N 2-A

K, 0.421 0.728 0.966 1 2.275 1.993 T, 10.52 10.93 11.20 12.5 8.59 14.83 T" 0.414 0.693 0.921 2 2.15 1.92

lated the closed-loop system with the derivative action, including a first-order filter with time constant equal to O.~T, , applied only to the measurement. This is a stand- ard practical implementation of PID control [7]. For each case, a setpoint change of magnitude 1.0 was introduced at t = 50 and a step load disturbance of magnitude 0.5 entered at t = 200. The process outputs are shown in Fig. 5 and the controller outputs appear in Fig. 6. Looking first at Fig. 6, it is very clear that these responses closely match the desired responses specified in eqn. 5, e.g. with a = 0.5, the initial change in the control signal for a unit setpoint change is close to 0.5 and the settling time of the control signal is approximately equal to 5 x T = 5 x 20 = 100. Fig. 5 shows that the speed of the process output response increases with increasing values of a but without any oscillations even with a = 1.5.

For comparison purposes, we have applied the design

IEE Proc-Control Theory Appl., Vol. 142, No. 4 , July 1995

methods of Ziegler and Nichols (Z-N) [l], Rivera et al. (IMC) [2] (Case F), and Zhuang and Atherton (Z-A) [3] to this example. The IMC method requires the choice of

-0 21 0 50 100 150 200 250 300 350 400

time

Fig. 5 Process output response for Example I a = 0.S a = 1.0 OL = 1.5

__ ~~~~

1 . 5 ~

0 0 50 100 150 200 250 300 350 400

time

Fig. 6 ~ DL = 0.5

a = 1.0 a = 1.5

Controller output response for Example I

- _ _ _ ~

the IMC filter time constant which we selected to be equal to 10. This performance specification is equivalent to a choice of a = 1 for our design. For the Z-A design, we have used their Table 7 for setpoint changes with the derivative action, including filter, in the feedback path. The process outputs are given in Fig. 7 and the controller outputs are shown in Fig. 8.

For the setpoint change, the initial change in the controller output is very large for the Z-N design (368%) and for the Z-A design (272%). In addition, the controller outputs for these two designs exhibit signifiant oscillations before reaching their final steady-state values. Both of these characteristics are undesirable in a process control context owing to modelling inaccuracies, satura- tion of final control elements, wear on final control elements, interactions with other process variables and operator distress [SI. The IMC design is much less aggressive, as expected from the performance specification. From looking at the corresponding process output responses, the Z-N and Z-A designs exhibit over- shoot, whereas the IMC design produces a monotonic rise to the new setpoint value. The load responses are

269

consistent in that the Z-N and Z-A designs produce faster recovery times than the IMC design and our designs owing to more aggressive changes in the controller outputs. Overall, the IMC design provides a level of performance which falls in between our designs of U = 1.0 and 1.5.

4.2 Example 2 Consider the problem of PID controller design for the following plant transfer function model :

4 0-

3 5 -

3 0 -

,a 2 5 - 3

3

L 2 0 -

E 1 5 -

1 0 -

0 5-

- -

8

e-sos G(s) = -

1os+ 1

For our design we have used a conservative choice for the parameter U of 0.25 and 5 = T/u = 40. This corresponds to a desired initial change in the control signal of 25% of its final value and an approximate settling time for the control signal of 160. The settling time of the closed-loop system (x) was estimated from the settling times of the control signal and the open-loop process to be 160 + 4 x 10 + 50 = 250. For this example we will not compare our results with the Ziegler-Nichols method because Astrom et al. [11] recommend against using this design method for processes of this type with delay to time constant ratios greater than unity. However, we will

1 4 r

-0 21 0 50 100 150 200 250 300 350 400

time

Fig. 7 _____ 2-A

2-N IMC

Process ouiput response for Example I

-~~~

OOL& 100 lb 200 2b 300 350 400 time

Fig. 8 ~ 2-A

2-N IMC

Controller output responsefor Example I

- ~ - ~

compare our results with the PID tuning rules proposed by Cohen and Coon (C-C) E121 as suggested by Astrom et al. [ll] to be a reasonable alternative in this case. We will again compare our results with the IMC-PID design proposed by Rivera et al. [2] (Case F) and with the tuning suggested by Zhuang and Atherton (%A) [3]. To make the comparison fair between our design and IMC, we will specify the performance level to be the same in both cases, i.e. the value for the filter time constant was selected to be equal to T = 40. This satisfies the condition for Case F that the filter time constant be greater than one-half of the process delay. The PID controller parameters for the four designs are summarised in Table 2. In each simulation experiment, a setpoint change of magnitude 1.0 was introduced at t = 50 and a step load disturbance of magnitude 0.5 entered at t = 500. The process outputs are shown in Fig. 9 and the controller outputs appear in Fig. 10. The results show that both the IMC and Z-A designs produce more aggressive

Table 2: PID peremetera for Example 2

a=0.25 IMC Cc 2-A

K , 0.275 0.538 0.517 0.508 7, 24.85 35.00 58.49 33.61 70 3.72 7.14 9.52 13.28

Fig. 9 __ a = 0.25

Process output response for Euunple 2

IMC C-c 2-A

_ _ _ _ . . . . . . .

Fig. 10 Controller output response for Example 2 - (I - 0.25

IMC C-C Z A

-. . . . . . . . . . . . .

270 IEE Proc.-Control Theory Appl., Vol. 142, No. 4, July 1995

responses to the setpoint and load changes than our design, and the C-C design produces a more sluggish response. However, our design provides a much smoother response than the other three methods in both the controller output and the process output with comparable performance. We found this observation to be true for these and many other examples (e.g. high-order systems, processes with right-half plane zeros) which we feel is directly related to our design philosophy. For instance, note that in Fig. 10, the response of the controller output follows almost exactly the specified trajectory for a value of a = 0.25 and T = 40 and it is this trajectory that produces the smooth process output response.

The models in eqns. 35 and 36 were used in different ways by the five design methods. In our method we used the model to calculate the process frequency response at the two frequencies o1 and oz. In the IMC and Cohen- Coon methods the transfer function model parameters were used directly in the respective formulas. For the Ziegler-Nichols and Zhuang-Atherton methods, the critical gain and period were calculated from the model.

5 Conclusions

This paper has presented a new PID controller design method which is based on limited frequency-domain process information. The algorithm uses a specification on the closed-loop control signals and has been shown, by way of two examples, to produce smooth behaviour in both the control signal and process output responses which is desirable in a process control context. The relationship between the closed-loop frequency response and the corresponding time-domain response has been

examined and used to choose the frequencies for controller design which have the greatest influence on the time- domain performance.

6 References

1 ZIEGLER, J.G., and NICHOLS, N.B.: ‘Optimum settings for automatic controllers’, Trans. A.S.M.E., 1942, 62, pp. 759-768

2 RIVERA, D.E., MORARI, M., and SKOGESTAD, S.: ‘Internal model control. 4. PID controller design’, Ind. Eng. Chem. Process Des. Den, 1986.25, pp. 252-265

3 ZHUANG, M., and ATHERTON, D.P.: ‘Automatic tuning of optimum PID controllers’, IEE Proc. D, 1993,140, pp. 216-224

4 ASTROM, K.J., and HAGGLUND, T.: ‘Automatic tuning of simple regulators with specifications on phase and amplitude marpins’, Automatica, 1984,ZO. pp. 64-651

5 HANG, C.C., ASTROM, K.J., and HO, W.K.: ‘Refinements of the Ziegler-Nichols tuning formula’, IEE Proc. D, 1991, 138, pp. 1 1 1-118

6 ASTROM, K.J.: ‘Assessment of achievable performance of simple feedback loops’, Int. J . Adapt. Control Signal Process., 1991, 5, pp. 3-19

7 ASTROM, K.J., nd HAGGLUND, T.: ‘Automatic tuning of PID controllers’ (Instrument Society of America, Rcscarch Triangle Park, N.C., 1988)

8 HARRIS, T.J., and TYREUS, B.D.: ‘Comments on “Internal model control. 4. PID controller design”’, Ind. Eng. Chem. Res., 1987, 26, pp. 2161-2162

9 BARNES, T.J.D., WANG, L., and CLUE’lT, W.R.: ‘A frequency domain design method for PID controller design’, Proceedings of American Control Conference, San Francisco, 1993, pp, 890-894

10 WANG, L., and CLUElT, W.R.: ‘A more efficient way to estimate step response cdicients using the FSF model‘, Proceedings of the American Control Conference, Baltimore, 1994, pp. 532-535

1 1 ASTROM, K.J., HANG, C.C., PERSSON, P., and HO, W.K.: ‘Towards intelligent PID control’, Automatica, 1992,w pp. 1-9

12 COHEN, G.H., and COON, G.A.: ’Theoretical consideration of retarded control’, Trans. A.S.M.E., 1953,75, pp. 827-834

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New frequency-domain design method for PID controllers