Ind. Eng. Chem. Process Des. Dev. 1962, 21 601-603
60 1
Parameter Estimation for the Second-Order-Plus-Dead-Time Model Chl-Tsung Huang and William C. Clements, Jr. Department of Chemical and Metallurgical Engineering, The University of Alabama, University, Alabama 35486
Some new, simple, and fast methods for estimation of second-order parameters, without computer usage, from step- and impulse-response data are suggested in this paper. Methods are presented for estimating the parameters from one point (inflection point), from two points, or from three points of the step-response data, using some new correlations. In addition, determination of second-orderparameters from the highest point of the impulse response, from numerical area integration, and from some regression equations is covered in this study. The methods are much easier to apply than the classical techniques usually used, and they provide better parameter estimates by eliminating reliance upon inflectional slope.
Introduction A mathematical model is usually required for the design of a control system. Since most real chemical processes are self-regulated, do not oscillate, and contain some amount of time delay, the second-order-plus-dead-time model has been used to approximate real systems for many years. Previous literature involving parameter estimation, without computer use, for second-order step-response data can be divided into two classes: (1)that which is based on inflection point and (2) that which is based on two points of the response curve. The classical method based on the inflection point of the step-response curve was developed by Oldenbourg and Sartorius (1948). Later, Sten (1970) modified this method by using a graphical technique to evaluate the two time constants, for the case of an overdamped second-order process. A method to estimate dead time based on the inflection point was also developed by Sten. Meyer et al. (1967) presented a graphical method by which the natural frequency and damping ratio can be determined from two points of the step-response curve. Recently, Sundaresan et al. (1978), from a theoretical basis, developed a graphical method to estimate all parameters, including dead time, from the step-response curve. To avoid the direct use of the inflection point, the method makes use of the first moment and the maximum slope. The second-order-plus-dead-time model normalized to a steady-state gain of 1 can be presented in terms of a single time constant and a damping ratio as
or in terms of two time constants as
Mathematically, these two forms are equivalent, but the first form is preferable in underdamped cases to avoid complex T values, and can be used for all T and 5. The second form holds only for the overdamped case. The unit step response for that case without dead time is 1 y ( t ) = 1 -(Tle-t/T1 - T2e-t/Tz) (3) T2 - T1 This form produces computational difficulties when a process approaches critical damping, as Tl then is near TP In other cases, one often fiids one large and one small time constant, especially when approximating a higher order system with a second-order model. The sensitivity of eq
+
0196-4305/82/1121-0601$01.25/0
3 to the time constants can cause difficult convergence using some least-squares algorithms. Another advantage of using the damping ratio form is that it could also apply (for 0.707 < f < 1)to a nonoscillatory system. Even though 5 < 1 is considered as an underdamped case, the oscillation is well damped and hardly perceptible in the range 0.707 < 4 < 1 for the time-domain step response curve. In the frequency domain, the amplitude ratio decreases monotonically with increasing frequency for f > 0.707; in this range, the amplitude ratio possesses no relative maximum. For 5 < 0.707, it is said that a resonance phenomenon occurs, and amplitude ratio does possess a relative maximum. Based on the above considerations, the damping-ratio form of the second-order-plus-dead-timemodel is chosen in this study. The aim of this paper is to present some methods for estimating T and 5 from step- and impulseresponse data that are faster and easier than techniques currently in use. Estimation of the model parameters is accomplished by a simple calculation that can be readily accomplished by a pocket calculator, so that a digital computer is not required. The estimated values can themselves be used as the final model parameters, or the methods discussed can be used to provide close initial guesses for nonlinear least-squares fitting, should it be desirable to use the latter approach to refine the estimates still further. Parameter Estimation Based on the Inflection Point of the Step Response Curve I t is obvious that one difference between a first-order and a higher order time-domain step response is that the higher order response has an inflection point on they vs. time graph. The inflection point seems to be a very important characteristic for the second-order step response. The relationship between the second-order parameters, T and 5, and the inflection point (tI,yI) can be shown from the theoretical step response equations to be (see Figure 1). (i) For 5 > 1
and vr=l-
0 1982 American Chemical Society
802
Ind. Eng. Chem.
Process Des. Dev., Vol. 21, No. 4,
1982
(ii) For 5 = 1
tr/T = 1
(6)
yI = 1 - 2e-' = 0.264
(7)
and (iii) For 5 C 1
mp1g
MI3
Figure 1. Damping ratio vs. yI and t I / T .
and
of 5, are given by Huang (1978). Inspection of the resulting data shows all the above ratios, t,/T, t 5 / T , ..., to be functions of 5 only. Now let =
cy
(tg -
tl)/t5 = f,(E)
and
8 = tg/h = f&) The inverse correlations, giving 5 as functions of a and 0,
From the above equations, it is found that y I and t I /T are functions of f only. Numerical data for yI and t I / T for the damping ratio, 5, ranging from 0.7 to 5.0 are given in Figure 1. Since the inverse function, 5 = f ( u I ) ,is not easy to obtain by an analytical method, a nonlinear least-squares curve fit to find the relationship between E and yI is necessary. An excellent empirical inverse function has been found
have been obtained by nonlinear least-squares curve fitting as
[ = 0.8637 y1-".578- 0.865
which has a usable range 1.5 I a I3.0 (0.7 IE I3.0), and
(S.E. = 2.62
X
(10)
(S.E.= 0.0142; r
4
r = 1.0)
This relation is valid for the range 0.0364 I yI I0.358 (ie., 0.7 If I 5.0). S.E. is standard error of estimate and r is multiple correlation coefficient. Using Sten's method (1970) to find dead time, the parameters of the second-order model can be estimated by the following procedures: (1)Locate the inflection point, (XI, yI), from the step-response curve. (2) Use Sten's method (1970) to find dead time, 0. (3) Calculate [ by eq 10. (4) Obtain t I ,where t I = Xi - 0. (5) Calculate T by the following equations
exp(6.4075~)+ 0 . 6 0 5 ~-~0.2
5 = 0.4646 X
= 0.13548 + 2.1586 X
(12)
= 0.9998)
exp(0.4620)
(13)
(S.E. = 0.0235; r = 0.9998) which has a usable range 5.2 I8 I 20.0 (0.7 I $, I 5.0). The following regression equations also have been obtained (Huang, 1978)
t,/T = 0.0137E2 + 0.072675 + 0.4445
(14)
(S.E. = 0.00237; r = 0.999928)
t,/T = 0.03922t2 + 1.096786 + 0.548 (S.E. = 0.0360; r = 0.99978) (15) and
tg/T = -0.0469E2 + 55 - 0.983 T = tI (for 6 = 1)
(11)
(16)
(S.E. = 0.0391, r = 0.99998) for 0.7 I 5 I5.0. Equations 14, 15, and 16 can be rewritten as
Parameter Estimation Based on Two or Three Points of the Step-Response Curve The drawback of the previous procedures is that an accurate inflection point is not easy to estimate from typical experimental data. More reliable methods which depend on two or three points of the step-response data will now be discussed. For a given value 5 in the pure second-order system, one can estimate the value of t / T (say t g / T )when the step response attains 90% of its final value, and estimate the value of t / T (say t2/T)when the step response has attained only 20% of its final value. It was found by Meyer et al. (1967) that the ratio of these two values [(t2/T ) / (tg/79 = t z / t g is ] independent of T. In our investigation, values of tl/T , t5/T , and te/ T for the range of 0.7 I 6 I5.0 have been calculated by spline interpolation. Details of these numerical data, with (ts- tl)/t5and tg/tlfor various values
tl Ttl = 0.013762 + 0.072675 + 0.4445 L5
Tt6=
0.03922f2+ 1.096785
Tt9=
+ 0.548
t9 -0.0469t2 + 55 - 0.983
(14a)
(15a)
(16a)
for 0.7 I 5 I 5.0. Using the above correlationswith Smith's method (1959) to find dead time, two techniques can be developed to estimate T and f from step-response data. (i) The Three-Point Method. (1)Find dead time by Smith's method (1959). (2) Obtain CY = (tg- t l ) / t Sfrom the step-response curve, where tl, t5,and tgare the time past the dead time at which the step response attains 1070,
Ind. Eng. Chem. Process Des.
a%,and 90% of its final value, respectively. (3) Calculate f by eq 12. (4) Calculate
T by eq 14a, 15a, and 16a, and
then take the average value. (ii) The Two-Point Method. (1)Find dead time by Smith's method (1959). (2) Obtain /3 = tg/tlfrom the step response curve. (3) Calculate f by eq 13. (4) Calculate T by eq 14a and 16a, and then take the average value. Parameter Estimation from Impulse Response: the Highest-Point Method Since the impulseresponse curve for the pure secondorder system monotonically increases before the peak time is reached, and ita maximum slope in this range occurs at the origin, it is easy to estimate dead time from impulseresponse data. If one draws a line tangent to the data curve, having a slope equal to the largest slope of the data between the origin and the peak of the impulse-response curve, the dead time can be taken as the time at which this tangent line intersects the time axis. This procedure effectively extrapolates the data to estimate where it would have begun rising from the time axis if it had done so at its maximum slope. From the point of view of the transfer function, one has (17)
sP(s)[step = Y(s) Iimpulse or
Also, the peak of the impulse-response curve occurs at the time where the step response has its greatest slope, i.e. the inflection point. It is therefore easy to estimate yI from the impulse response data by numerical integration as y(xI)lsbp y1 =
-
S,X;(t)limpube d t (18)
Y(")lstep JmY(t)limpde
dt
The correlations of eq 10 and 11 would then be used to estimate T and f . Further Comments From the authors' experience, the three-point method seems to be better than the two-point method, which is not surprising since more points are taken. However, the ratio ( t g- t,)/t5 for a pure second-order system is about 3.1 and is almost constant in the range 3.0 < f < 5.0. It is very difficult to get an accurate f from the ratio, a,in this range. The two-point method is therefore useful to complement the three-point method for this range of 5. If the estimated damping ratio is very great, say 5 > 5.0, this means that the inflection point occurs at a time at which the response is less than 3.6% of its final steadystate value. The inflection point therefore is barely perceptible. In this case, a first-order-plus-dead-time model is suggested. For some nonoscillatory processes, the inflectional slope of the step-response curve is very steep, and to fit data of
Dev., Vol. 21, No. 4, 1982
603
this type to the second-order-plus-dead-time model may be difficult. Huang (1978) used a nonlinear least-squares technique to fit the data computed from the axial dispersion model (Brenner, 1962) to the second-order-plusdead-time model. From his results, damping ratios near 0.8 were consistently found for Peclet numbers in the range 10 to 160. Accordingly, a rule of thumb to treat a stepresponse curve having very steep slope at the inflection point is also suggested here: one can let 5 = 0.8, and solve for T by the methods suggested earlier. Since 5 is set at 0.8, eq 14a, 15a, and 16a then become Tt, = 1.9474t1 (14b)
Tt, = 0.6646t5
(15b)
Ttg= O.3354tg
(16b)
The techniques summarized here will allow the experimenter to rapidly estimate parameters for the model from his data. Due to the accuracy of the simple algebraic correlations, the estimates should be about as good as could be obtained from a nonlinear regression, yet are explicit in 5 and require no computer usage. Nomenclature G(s) = system transfer function r = multiple correlation coefficient s = Laplace transform variable S.E.= standard error of estimate t = time t I = the time past the dead time at which the inflection point occurs, = XI - 0 tl, t5, t g = the time past the dead time at which the step response attains lo%, 50%, or 90% of its final value, respectively T , T,, T2 = time constants of the second-order model !!'I, Tt,, Ttg= estimated time constants X ( s ) = Laplace-domain forcing function XI = time of the inflection point in step response or the highest point in impulse response P(s) = Laplace domain system response y ( t ) = time domain system response yI = fractional step response at the inflection point, (response at inflection)/y(a) Greek Letters = (t9 - t l ) / t 5 P = t9/tl $, = damping ratio 0 = dead time
Literature Cited Brenner, H. Chem. Eng. Sci. 1962, 17, 229. Huang, C.-T. M.S. Thesis, The University of Alabama, University, AL, 1978. Meyer, J. R.; Whitehouse, G. D.; Smith, C. L.; Murrill, P. W. Instrum. Contra/ Syst. 1967, 40(12), 76. OMenbourg, R. C.; Sartorius, H. "The Dynamics of Automatic Controls"; The American Society of Mechanical Engineers: New York, 1948; pp 75-79. Smith, 0. J. M. ISA J . 1959, 6(2), 28. Sten, J. W. Insfrum. Techno/. 1970, 17(9), 39. Sundaresan, K. R.; Prasad, C. C.; Krishnaswamy, P. R. Ind. Eng. Chem. Process D e s . D e v . 1970, 17, 237.
Received for review December 11, 1980 Accepted April 12, 1982
