3052
Znd. Eng. Chem. Res. 1995,34, 3052-3058
Closed-LoopIdentification by Use of Single-valued Nonlinearities Mats Friman and Kurt V. Waller* Process Control Laboratory, Department of Chemical Engineering, Abo Akademi, FZN-20500 Abo, Finland
Closed-loop identification of the ultimate gain (K,) and critical frequency (w,) of a feedback control loop is studied. The accuracy of the ktrom-Hagglund autotuner is discussed, and a simple method of improving the accuracy of the commonly used procedure is proposed and tested by simulation. A modification of the method where the traditional one-parameter relay is replaced by a two-parameter nonlinearity is also proposed. Because the selection of the two parameters of such a nonlinearity is nontrivial, a n iterative procedure for the identification of K, and w, is proposed. The procedure has been found to be convergent and to give reliable and accurate identifications in a number of examples typical of chemical engineering.
Introduction The Astrom and Hagglund (1984) relay feedback autotuner has been widely accepted for controller tuning for single input-single output (SISO) systems (8strom and Hagglund, 1988; Dumont et al., 1989; 8strom et al., 1993a) as well as for multiple input-multiple output (MIMO) systems (Loh et al., 1993; Palmor et al., 1993; Shen and Yu, 1994; Friman and Waller, 1994). Relay feedback has also proved to be useful and efficient for closed-loop identification of process transfer functions (Luyben, 1987; Li et al., 1991; Chang et al., 1992). Even systems that traditionally have been difficult to control and tune have successfully been implemented by use of relay feedback. One example is the tuning of nonlinear pH systems reported by Lee et al. (1993). Several commercial controllers available today are equipped with relay autotuning (Astrom et al., 1993b). The advantages of the relay feedback autotuner are obvious. Only some elementary a priori knowledge of the process is needed, and the identification and tuning sequences are easy to automate and easy to use. Since it is a closed-loop test with bounded input amplitude, the output can be kept close t o its setpoint during identification. A number of modifications and extensions have been propose$ since the launch of the single input-single output Astrom-Hagglund autotuner in 1984. Luyben (1987) demonstrates the advantages of the autotuner for identification of multivariable transfer functions for distillation column control. The identification method was later extended by Li et al. (1991). Their basic idea is to connect additional dead time to the process in order to identify several points on the Nyquist curve, something that was earlier suggested by Astrom and Hagglund (1984). This often gives more equations for calculation parameters than there are unknown parameters, so that, e.g., a least-squares method can be used to determine the transfer function parameters. Chang et al. (1992)derive analytic expressions for the period of oscillation and amplitude ratios for simple transfer functions. By using these expressions in the calculation of the unknown parameters, the error in the identification can be significantly reduced for dead time systems. The derivation of transfer functions in general will not be treated in detail in this paper; we shall mainly focus on the identification of K, and mu. The dead time L can often be read from the initial part of the outputs during relay feedback start-up, and the steady state gain K can be calculated from a step change in the input 0888-5885/95/2634-3052$09.00/0
30
0
Time
Figure 1. Process output y and input u for relay feedback identification of the system G(s) = exp(-s)/s + exp(-4.5s). The unsuccessful multiswitch limit cycle relay identification (solid lines) is plotted together with the successful ideal saturation (with gain k = 1)identification (dashed lines).
signal as described by Luyben (1987). Steady-state-gain identification for multivariable systems is also discussed by Pensar and Waller (1993). When these four parameters (K,, mu, L , and K ) are known, four parameters in a transfer function can be determined. The selection of process transfer functions is discussed in, e.g., Li et al. (1991). The relay has proven to be a useful tool in process identification and autotuning, and a large number of successful applications have been reported in the literature. There are, however, situations where the relay is not suitable for process identification. A number of results where multiswitch limit cycles and even chaotic situations have occurred with relay feedback have been reported (Holmberg, 1991; Billings et al., 1984). One such example is when the system described by the transfer function
is connected to relay feedback. A simulation of this is shown in Figure 1. A multiswitch limit cycle occurs, and this system can clearly not be identified by the use of a relay autotuner. Note that the system in eq 1does not have multiple frequencies with a -180" phase lag. We have found that this problem arises more frequently with an increasing number of nonlinearities in the system, e.g., with several simultaneously switching relays that are connected in feedback loops, something recently suggested for autotuning of MIMO systems (Palmor et al., 1993; Loh and Vasnani, 1994). Also in Figure 1, we have simulated the process given by eq 1,
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 3053 0.15s
Y )
Figure 2. Setup of a nonlinear feedback experiment.
but with the relay feedback replaced by another nonlinear feedback discussed later in this paper. Then a successful identification of the parameters Ku and w, can be made. In addition to situations like the one above where no result is obtained by use of relay identification, there are situations where the standard method can be quite inaccurate. One example is a process dominated by dead time, a situation not uncommon in the process industries. For such processes the filtering of higher frequencies is poor. The common approach, i.e., to use the maximum amplitude values of the output instead of the output of the first harmonic, can then lead to significant errors in the identified maximum gain in the feedback loop. This is illustrated in the following section. It should be emphasized that there are many situations where one does not know whether the results obtained by the standard procedure are accurate or not. The present paper discusses how the standard relay identification procedure of h t r o m and Hagglund (1984) can, by quite simple means, be improved, albeit a t the expense of some added complexity. The modifications suggested may significantly increase the accuracy of the identification and permit a successful identification of Ku and wu for systems like the one described by eq 1.
Relay Identification The structure of the hrom-Hagglund autotuner (AW) is shown in Figure 2. It consists of an ideal relay n connected as a feedback controller to a stable process. The nonlinear function for the relays used in autotuners is given by
h ezO {-h e < O where h is the relay amplitude. This system starts oscillating if the process has a phase lag of at least -n radians (-180”), which is true for all real processes. If a is the amplitude of the first harmonic of the output oscillations, then according t o the describing function analysis, the relay will play the role of a gain-varying proportional controller. This gain is given by (Atherton, 1982)
u(e)=
4h N ( a )= na
(3)
Because the phase lag between the process input and output is -n radians, we have a situation similar to the Ziegler-Nichols (1942) ultimate sensitivity experiment and consequently,
K = -4h na
2n
w, = -
P
(4b)
Here P denotes the period of the limit cycle. The information about the process obtained from this experiment is often appropriate in tuning a simple feedback controller.
0
30
Time
Figure 3. Process output y and input u for relay feedback 1). identification of the system G(s)= (2.5s 1)exp(-5s)/((50s (0.1s 1)).The experimental signals (solid lines) are plotted together with the first harmonics (dashed lines). The definitions of the amplitudes a , a,, and h are also included.
+
+
+
We shall first discuss the accuracy of eqs 4a and 4b. From the limit cycle of the relay feedback system we notice that the input to the process u is a square wave and the phase lag between process input and output is -n radians. If we replace the square wave with a sinusoid of the same frequency as the relay, the frequency components of the input signal are no longer the same and the phase lag between input and output is not generally the same -nradians. Thus, the identified critical frequency in eq 4b usually is only an approximation of the critical frequency, the latter generally defined for a sinusoid input signal. Consequently, if the frequency of the relay differs from the critical frequency of the system, the ultimate gain in eq 4a will be based on the wrong frequency and, thus, only an approximation as well. We will illustrate the discussion above with the following example. Connect the system
+w 5 s + 1)(50s + 1)
(2.5s
G(s)= (0.1s
to relay feedback in order to identify the ultimate gain and critical frequency. The output y and input u from the relay feedback simulation are shown in Figure 3 together with the first harmonics of y and u. We determined the first harmonic of the output in the following way: from the autotuning experiment we registered the frequency of the relay. The input u in a second simulation was a sinusoid with the same frequency. The output signal for that simulation is then what is defined as the first harmonic of y. If we look a t the points where y crosses zero, we notice that the measured output y crosses zero before the first harmonic of y. This is an illustration of the fact that the phase lag caused by higher harmonics affects the identification and that the frequency given by the relay oscillation (in this example) is higher than the critical frequency of the system. For this particular system the difference is significant, the critical frequency of the system in eq 5 (i.e., the frequency of the sinusoid that would give a phase lag of -n radians) is w, = 0.49 radk but the relay switches with the frequency w = 0.59 rads, a difference of FO%. Astrom and Hagglund (1984) discuss the determination of the amplitude and period from the oscillation in the following way: “The period of an oscillation can easily be determined by measuring the times between zero crossings. The amplitude may be determined by measuring peak-to-peak values of the output. These
3054 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995
estimation methods are easy to implement because they are based on counting and comparisons only. Since the describing function analysis is based on the first harmonic of the oscillation, the simple estimation techniques require that the first harmonic dominates. If this is not the case, it may be necessary to filter the signal before measuring. More elaborate estimation schemes like least squares estimation and extended Kalman filtering may also be used to determine the amplitude and the frequency of the limit cycle oscillation. Simulations and experiments on industrial processes have indicated that little is gained in practice by using more sophisticated methods for determining the amplitude and period." However, as will be shown below it may be important that the variable a in eq 4a is interpreted as the amplitude of the first harmonic and not as the amplitude given by peak-to-peak measurements of the output.
The Importance of the First Harmonic The identification of process transfer functions with relay feedback has recently been studied in the literature (Li et al., 1991; Chang et al., 1992). These authors assume that from the relay feedback experiment only peak-to-peak measurements of the output oscillations are available, not the first harmonic. This approach can sometimes be useful, but if we want to use eq 4a to calculate the ultimate gain, this approach requires the assumption that the amplitude given by peak-to-peak measurements is equal to the amplitude of the first harmonic. The following example shows that we get different results depending on how we interpret the variable a. It also shows that there are cases where this difference is significant. Consider the first-order-plus-dead-time system
e -Ls G(s)= Ts 1
+
(6)
connected in a relay feedback loop. We simulated this system for different values of the ratio T I L and calculated the ultimate gain according to eq 4a in the following two ways: (1)we interpreted the variable a as the peak-to-peak amplitude of the output oscillations (i.e., the common way), and (2) we interpreted a as the amplitude of the first harmonic in the process output. We calculated the first harmonic from the Fourier series coefficients by choosing a time interval t = [to t11, corresponding to a whole number n of limit cycles, from the process output y so thaty(t0) = y(td = 0. If the relay oscillations are symmetric (i.e., the relay switches a t a constant frequency) and if the system is linear, the cosine Coefficients of the Fourier series are equal to zero and the amplitude of the first harmonic is calculated from the correlation between y and the sine wave of period (tl - tolln:
In this paper we denote the amplitude by a, when we mean the amplitude of the process output calculated from the minimum and maximum points of the oscillation, and by a when we mean its first harmonic (see Figure 3 for an illustration). Figure 4 shows the results of the identification for the two techniques. Figure 4 also shows that for a first-order-plus-dead-time system, the identification assuming a = a, gives an error in K,
0.2
E
.o 1
L,d 001
,,,.,,, .I .,.,,,ul 01
J
,,.,,,~.,,,~,~~,,,,,~, 10
1
100
1000
TIL
Figure4. Relative error in autotune identification of a first-orderplus-dead-time system using an ideal relay. Standard ATV identification ( x ) and the identification according to the FH method (+I.
of almost +30% for systems dominated by dead time and of almost -20% when the system is dominated by the time constant. In the next section the identification of K, according to eq 4a, where a is interpreted as the first harmonic calculated by eq 7, is referred to as the FH method (for first harmonic). In Figure 4 it is also interesting to note the connection between the error in the critical frequency and the error in the ultimate gain. If the frequency is exactly identified, the calculation of K, according to eq 7 is exact for linear systems. It is also interesting to notice that the peak of error in the estimation of wu is obtained for the ratio TIL approximately giving the minimum error in estimating K, with the standard ATV method. This is, however, a pure coincidence.
Identification Using Two-Parameter Nonlinearities The FH method discussed in the previous section treats only the amplitude. Thus, no improvement of the identification of the critical frequency is obtained by this method. Figure 4 shows that the error in the relay identification of the critical frequency is not larger than about 5% for a first-order-plus-dead-time system, which is much less than the maximum error in the ultimate gain. The example above where the critical frequency of the process given by eq 5 was identified with an error of 20% shows that there are processes where the error can be considerably larger than shown in Figure 4. There may thus be a certain incentive for improvement also of the identification of the critical frequency. An improvement is obtained by use of a suitable twoparameter odd single-valued nonlinearity (SVNL) instead of the relay in the feedback loop. An odd singlevalued nonlinearity is simply a unique nonlinear function u = n(e) with odd symmetry, i.e., n(e) = 4 - e ) . It is obvious that it takes less computing efforts to determine the amplitude given by the minimum and maximum points from the limit cycle oscillations than it does to determine it by the computation of the first harmonic. We shall therefore study the identification for these two cases separately. Consider the system in Figure 2. Assume that the SVNL denoted by n causes the system to oscillate with the output amplitude a of the first harmonic. The describing function of the nonlinearity and the ultimate gain for the process, are then given by (Atherton, 1982 p 81)
Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 3055
I - -7-
This expression is valid for all nonlinearities. For the relay it simplifies to eq 4a. The integral cannot always be solved analytically, but it can be solved numerically or by the approximation given by Atherton (1982 p 97)
I
+ n(al2)+ &n(a&l2))
%(a) 3a
KIJ
(9)
Figure 5. Ideal saturation.
The identification sequence is illustrated by the following example. Let the nonlinear element n in Figure 2 be given by an ideal saturation, defined by
e > hlk -hlk 5 e 5 hlk e < -hlk
[
u(e) = ke
:h
(10) Ideal saturation is illustrated in Figure 5. Using eq 8, the ultimate gain for an oscillating system, connected to ideal saturation feedback with process output amplitude a of the first harmonic, is given by (Atherton, 1982 p. 85)
k K , =-(2a x
+ sin 2a);
? lil
-0 1 001
10
01
100
1000
TIL
Figure 6. Relative error for ideal saturation identification. ( x ) Standard ATV, (0) ideal saturation, (+) ideal saturation in combination with the FH method.
a = arcsin
Note that the nonlinearity has two parameters which have to be chosen by the user, the height h and the gain k. The height h should be selected on the same basis as the height of the relay. The gain 12 must be selected large enough to ensure limit cycles. With this constraint fulfilled a smaller k generally gives a more accurate identification. The improvement is increased if amis used as an approximation for a in the equation above. The limit can be found in a few steps by trial and error. A systematic method is t o start with k = 00 (which is the same as an ideal relay) in order to get an estimation of the ultimate gain K u , e s t . The gain k in the next feedback identification is then based on this value, e.g., by using k = K S u , e s t . The parameters obtained from this step will then be more accurate than the parameters obtained from a standard ATV test. Here ke expresses a “safety margin” which has to be determined by the user. If the relative error in the ideal relay identification for the gain K, is (12) the oscillation condition k > K, implies i
1
ke > + eKu
(13)
Looking a t the relative error for the first-order-plusdead-time system (Figure 41, and according to our experience (based mainly on work with first-order-plusdead-time systems and second-order-plus-dead-time systems), k, should preferably be chosen in the range 1.5-2.0. To demonstrate the identification by way of using the ideal saturation nonlinearity we simulated the process given by eq 6 for a number of different time constant/ dead time ratios TIL,connected t o ideal saturation feedback. As concluded above, two parameters of the
hlk
Figure 7. Nonlinearities used in identification experiments. (a) inverse tangent, (b) sigmoid, and (c) exponential saturation.
nonlinearity are required for the identification and therefore we made two different identification experiments. 1. The first one was a standard ATV identification (i.e., k = 00) which gives an estimate of the ultimate gain K u , e s t . The estimate helps us to get an appropriate gain for the ideal saturation identification in the next step. Because we only need a crude estimation of the ultimate gain here, we calculated Ku,est according to the standard ATV method, not the FH method. 2. The second experiment was an ideal saturation identification with a gain k calculated according to k = =,,est where Ku,estis the ultimate gain identified in the first step. The ideal saturation identification gives a more accurate identification of the ultimate gain and the critical frequency than the relay-based ATV method does. For this experiment we calculated the ultimate gain using eq 11in the two different ways described in this paper: (a) we assumed that a = am (denoted as ideal saturation in Figure 61, and (b) we calculated the amplitude of the first harmonic a by eq 7 (denoted as ideal saturation in combination with the FH method). The results are shown in Figure 6, and we can notice that the chosen modifications of the ATV method have brought about significant improvements in the accuracy. There are a number of other suitable nonlinearities that can be used as a feedback controller in the identification of the critical point on the Nyquist curve. Some examples are given in Table 1and Figure 7. (The nonlinearities used are shown only for the positive e-axis, all nonlinearities which we consider are odd functions, n(-e) = -n(e).)
3066 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995 ' k L E
02r-j 00
e
.02
sL3"
x x x
x x x
e
L
li
w
-01
oob ;,;,,; ;,X*&-j
;
001[0F7---]
01
001
100
10
1
1000
.o
1
001
01
1
10
100
1000
TIL
Figure 10. Same as Figure 6 but for sigmoid nonlinearity. -Frk 02
ka
E
'
-
. O O
00-+ +
'X""
.-'.I
.
' ,..."I '
x x
x x x
'
-"I
.' .*
O o X
--x
+ +:
L
42Lu m7
01
,....,
,
x x x
. .... .
l...'.l r 1...11.,
1-91
a
OO-"
"-
L
e
Lu
x: .I.....
.
3"
t
-
.
-....-'
.01
L
' .U ' S.'
Lu
-01
001
01
10
1
100
1000
TIL
Figure 11. Same as Figure 6 but for the second-order-plus-deadtime system G(s) = exp(-Lsj/(Ts 1)2.
+
calculation of K,
type of nonlinearity
*(A
relay
4h -
nag g
w
na
01
ideal saturation
h e > hlk ke -hlk c e 5 hlk -h e < -h/k
k
+ sin 2aj; where
$-2a
a = arcsin(&); TIL
51
inverse tangent
2h
dX-1 a2k
exponential saturation
h(1 - e u(e) =
sigmoid
2 2ke 1+ e X P ( T )
)
eq 9
We simulated the identification Drocedure for the different nonlinearities given in Tadle 1 and Figure 7 for the same first-order-plus-dead-time process as in eq 6 using the same sequences and the same parameter k, = 2 as for the ideal saturation identification described above. For these nonlinearities we used eq 9 for the calculation of the ultimate gain Ku,except for the inverse tangent where the integral in eq 8 can be solved analytically (see Table 1for the calculation of K,). The amplitude a of the first harmonic needed in the calculation of K, was obtained from the output amplitude a, by numerical integration of eq 7. The results of the identifications are shown in Figures 8-10. The figures show that the accuracy of the identification may be
Figure 12. Same as Figure 6 but for the second-order-plus-deadtime system G(sj = exp(-Lsj/((Ts + l)(q'Ts 1)).
+
significantly improved by use of other single-valued nonlinearities than the ideal relay, albeit at the expense of some added complexity of the procedure. Figures 11and 12 illustrate how the procedures work for some second-order-plus-dead-time systems. A notable difference compared to first-order-plus-dead-time systems is that the error in the identification of the critical frequency can also be negative. In this section we have claimed that, for the twoparameter nonlinearities treated, the smaller the gain k is (provided it is large enough to cause continuous oscillations), the better the identification of K, and w,. To understand this, consider the identification of the system given by the transfer function in eq 5. A simulation for different values of k for the ideal saturation identification is shown in Figure 13. From the simulations we see that as k decreases, we get smoother, more sinusoid-like oscillations in the input u. As concluded above, one of the major error sources of relay feedback is that higher harmonics are not always properly eliminated. It is therefore likely that these smooth oscillations give a better identification. We also notice that with a smaller gain, the controller will operate more like a proportional controller and if we decrease k until we reach the ultimate gain k = K, the ideal saturation experiment transforms into the Ziegler-Nichols ultimate sensitivity method.
Ind. Eng. Chem. Res., Vol. 34,No. 9, 1995 3057
I
.
0
0
2
30
Time
Figure 13. Process output y and input u for ideal saturation identification of the system G(s) = (2.5s + 1) exp(-5s)/((O.ls 1)(50s 1)).The parameters are as follows: height h = 1 (in all cases), gain k = m (solid line), k = L5K, (dashed line), k = l.lKu (dotted line) where K, is the ultimate gain of G(s).
+
+
Why then not use a proportional controller and tune it to the verge of instability as suggested by Ziegler and Nichols already in 1942? The answer is that the Ziegler-Nichols method is difficult to automate and that it is difficult to adjust the amplitude of the input oscillations. The smoothness of the input oscillations also explains the success of the identification of the process described by eq 1 and shown in Figure 1.
An Iterative Procedure On the basis of the discussion in the previous section an iterative procedure for identification of the critical point on the Nyquist curve can be suggested. It can be observed that the limit cycles in the ideal saturation feedback experiment have the following useful feature: either the process output amplitude a m is larger than h/k or there is no limit cycle at all. The response thus determines whether the gain k is larger or smaller than K,, and after that a systematic search for the critical point on the Nyquist curve can be made. Such an algorithm is given below. Algorithm for iterative identification of ultimate gain and critical frequency: 0. Select the relay amplitude h. 1. Perform a relay feedback experiment, giving a crude estimate of the ultimate gain K,J. Set n = 1 and k = Ku,l. 2. Set n = n 1 and make an ideal saturation feedback experiment. Register the amplitude of the process output a m . 3. If a , Ih/k (stable system with no limit cycles), (a) a lower bound of K, is found: Ku,min = k . 4. If a , > h / k , (oscillating system with stable limit cycles) (a) an upper bound of K,, is found: Kqmax = k . (b) Calculate the nth estimate of the ultimate gain Ku,nby eq 11. (c) If no further improvements have been made in the identification of K,, register the ultimate gain and critical frequency, go to step 8. 5 . If Ku,,in is not yet found, set k = K,/k,, go to step 2. 6 . If K,,,,,, is not yet found, set k = k&,, go t o step 2. 7. Set k = 0.5(Ku,min K,,,,,,), go to step 2. 8. End
+
+
6
4
8
iteration #
Figure 14. Relative error for the steps in the suggested iterative procedure. The process used is the first-order-plus-dead-time system described by eq 6 with the following ratios between time constants and dead times ( x ) TIL = 0.01; (0)TIL = 1;and (+) TIL = 100. 0.3
k
-0.3 I
0.1
.-
e
-O0 1
/+
'
I
'
I
o
.
1
o
k
E
iteration #
Figure 15. Same as Figure 14 but for the second-order-plus-deadtime system G(s) = exp(-Ls)I(Ts + 1)*.
The constant k , in steps 6 and 7 is required in the procedure. It must be greater than one in order to make the algorithm converge. We have found appropriate values to be in the range 1.1-2.0. The identification of the first-order process given by eq 6 as well as some second-order processes was tested through simulation with the algorithm described above and with k , = 1.5. The results are shown in Figures 14 and 15. Note that there are iterations where no identification has been made. These are the steps where the closed loop has become stable and no limit cycles have occurred. The identification of different processes through the iterative procedure described in this section has also been tested for the process given by eq 5 and for the inverse-response process given by
G(s) =
-s+
20s
1
-8
+ le
and the fourth-order process
(15) The results of the identifications are shown in Figure 16. Note that the fourth-order process in eq 15 behaves like a low-pass filter and well attenuates the higher harmonics. The relay seems t o be suitable for identification of such processes, whereas processes dominated by nonminimum phase characteristics could seemingly benefit from the methods suggested in this paper. We have chosen to use the amplitude a m given by peak-to-peak measurements in the iterative procedure above. This might at first sound a bit confusing taking into account the discussion where we showed the importance of calculating the first harmonic to get proper results in the identifications. However, for
3058 Ind. Eng. Chem. Res., Vol. 34, No. 9, 1995
Literature Cited
e -03 01
-011
'
0
'
2
'
'
'
4
'
6
'
'
8
I
iteration #
Figure 16. Same as Figure 14 but for process given by eq 5 ( x ), the inverse response process given by eq 14 (0)and the fourthorder process given by eq 15 (+).
transfer functions describing typical chemical engineering processes, we have found the algorithm to be so rapidly converging that the job of data processing and numerical integrations that the calculation of a requires is not justified.
Summary and Conclusions The use of different nonlinear elements for closedloop identification of the critical point on the Nyquist curve has been treated. If a relay is used as a feedback controller during the identification, significant errors in the identification of the ultimate gain K, may be obtained if the amplitude given by the maximum and minimum points in the output oscillations is used instead of the amplitude of the first harmonic. The accuracy of the identification of the critical frequency and thus also of the ultimate gain may be improved if suitable two-parameter nonlinearities are used instead of the relay. The largest improvements have been found for processes dominated by nonminimum phase characteristics such as dead time and RHP zeros. An iterative procedure is suggested for accurate identification of the ultimate gain and the critical frequency. The identification is similar to a standard ATV identification, but the relay in the feedback loop is replaced by an ideal saturation where one of the two parameters describing the ideal saturation is iteratively obtained. The procedures are illustrated by simulation of firstorder-plus-dead-time as well as of second-order-plusdead-time systems for a large range of ratios between dead time and time constant. Also studied are inverseresponse and higher-order systems.
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Received for review November 23, 1994 Revised manuscript received May 17, 1995 Accepted May 25, 1995@
Acknowledgment The results reported have been obtained during a long-range project on multivariable process control supported by Tekes, The Academy of Finland, Nordisk Industrifond, Neste Foundation, and Neste Oy. This support is gratefully acknowledged.
I39406963
Abstract published in Advance ACS Abstracts, August 1, 1995. @
