GREEK LETTERS = number of sampling periods the forward signal is
a
6
e ui
$
6
O
o,,a,,, +p
delayed = vector of constants for constraints on K ( t ) = measured temperature = ith root to be canceled = a load function = load function vector = matrix defined in Equation 17 = weighting function vector = weighting functions defined in Equation 29 = load function vector = & j.l
+
SUBSCRIPT s = based on n samples n = based on I samples 1 SUPERSCRIPTS * = a sampled variable = a vector or matrix - = see Equation 30
,.
+,
Hadley, G., “Linear Programming,” Addison-Wesley, Reading, Mass.. 1962. ~Ijirii-Y,, Carnegie Institute of Technology, GSIA Library Program No. 1004 (1961). Kermode, R. I., Stevens, \Y. F., Can. J . Chem. Eng. 39, 81 (1961). 5,396 (1966a). Koppel, L. B., IND.ENG.CHEM.FUNDAMENTALS Koppel, L. B., ZSA Journal 13 (lo), 52 (1966b). Kuo, B. C., “Analysis and Synthesis of Sampled-Data Control Systems,” Prentice-Hall, Englewood Cliffs, N. J., 1963. Min, H. S., Williams, T. J., Chem. Eng. Progr. Symp. Ser. 57 (36), 100 (1961). Mosler, H. A., Koppel, L. B., Cou hanowr, D. R., IND.ENG. 5 , 297 (1966g. CHEM.FUNDAMENTALS Porcelli, G., Fegley, K. A., Joint Automatic Control Conference, Stanford University, p. 412, 1964. Propoi, A. I., Automation Remote Control 24 (7), 837 (1963). Ragazzini, T. R., Franklin, G. F., “Sampled-Data Control Systems.” McGraw-Hill. New York. 1958. Tong,”. C., J . Franklin Znst. 278,28 (1964). Tou, J. T., “Digital and Sampled-Data Control Systems,” McGraw-Hill, New York, 1959. Whalen. B. H., I R E Trans. Auto. Control A C 7 (4), 46 (1962). Willianls,T. J., Min, H. S., I S A Journal 6 (9), 89 (1959). Zadeh, L. A., I R E Trans. Auto. Control A C 7 (4), 45 (1962). I
~
literature Cited
Fegley, K. A., I R E Trans. Applications Industry 83 (72), 188 (1964). Fegley, K. A,, Hsu, M. I., I R E Trans. Auto. Control V.AC10, 114 (1965).
RECEIVED for review December 8, 1966 ACCEPTEDJune 27, 1967
INVARIANT IMBEDDING, NONLINEAR FILTERING, AND PARAMETER ESTIMATION E. STANLEY
LEE’
Research and Development Department, Phillips Petroleum Co., Bartlesville, Okla.
The invariant imbedding concept is used to derive sequential estimator equations in the theory of nonlinear filtering and estimation. The reaction rate constant and the concentration of the reactant are estimated by these estimator equations. The least squares criterion is used.
HE
estimation of parameters is of interest not only in adap-
T tive control, but also in other areas where the mathe-
matical representation of the process from various experimental data is to be determined. I n a previous paper (Lee, 1968a), quasilinearization technique was used to estimate these parameters. I n this paper, the invariant imbedding concept (Bellman and Kalaba, 1960; Lee, 1968b; Wing, 1962) is used to derive some useful estimator equations for nonlinear dynamic systems. No statistical assumptions are made concerning the disturbances or measurement errors. For most practical problems, the determination of valid statistical data concerning these disturbances is a difficult problem in itself. T h e generally used least squares criterion is used to obtain the estimator equations. I n contrast to the nonsequential estimation scheme resulting from the usual classical approach, the estimator equations obtained by invariant imbedding are sequential estimators. A typical nonsequential estimation scheme is discussed by Lee (1968a).
Present address, Department of Chemical Engineering, Kansas State University, Manhattan, Kan. 1
164
I&EC FUNDAMENTALS
The sequential estimation scheme has two advantages over the nonsequential ones for dynamic systems. First, for nonsequential estimation schemes, each time additional observations or measurements are to be included the entire calculations must be repeated. Secondly, because of these repeated calculations, nonsequential estimation schemes are much more difficult to implement in real time than sequential ones. The nonlinear estimation problem treated in the present work is essentially an extension of the well-known linear problem treated by Kalman and Bucy (1961). T h e estimator equations were originally obtained by Bellman and coworkers (1964). Estimation Problem
Consider a system whose dynamic behavior can be represented by the nonlinear differential equation
The state of the system, x , is being measured or observed starting a t an initial time t o = 0 and continues to the present time, t,. Because of the presence of noises or measurement
system represented by Equations 8 and 9 where
errors, the observed state, z, of the system does not represent the true state. Let
r(t) =
x(t)
+ (measurement or observation errors)
(2)
Based on this observed signal ~ ( tin) the interval 0 5 t 5 t,, an estimate of the present state x(t,), a past state, x ( t J , or a future state, x ( t 2 ) may be sought. Let us first consider the problem of estimating the present or current state, x ( t r ) , of the system. T h e other two problrms can be solved in the same way. I n fact, the entire trajectory or profile is estimated during the estimation of x(t,). Using the classical least squares criterion, the problem is to estimate the current state of the system a t t,, such that the following integral is minimized
J
=
- z(t)12dt
[x(t)
(3)
where z ( t ) is the observed function. Function x ( t ) is detert t, by differential Equation 1. mined on the interval 0 This problem can be stated differently as follows: Based on the observation r ( t ) , 0 t 5 t,, estimate the unknown condition
< <
