Parameter estimation is the determination of coefficients in a theoretical model by interpretation of experimental results. In general, there are two classes of parameter estimation:

Class 1. Parameters are determined that are required only to cause theoretical results predicted by the model to agree with experimental results. This class of parameter estimation is related to the problem of constructing an empirical model, but in this case the problem consists of assigning coefficients in a model with a predetermined structure. Also, coefficients that are adequately known can be fixed.

Class 2. Parameters are determined that accomplish the objectives of class-1 parameter identification; but, furthermore, the parameters must be “physical.”

The reason for this distinction is that there may be many sets of coefficients that give equally good agreement between theory and experiment, but there is only one set that would agree with the coefficients that would be deter­mined individually in independent, perfect measurements. Clearly a method that will solve class-2 problems will also solve class-1 problems.

One form of parameter estimation uses the experimental response (either frequency response or time response) at N selected points. The parameter estimation procedure finds the set of coefficients that minimizes the error between theory and experiment. The error is defined as follows:

where Уе(/с) is the experimental response at observation point /с, and Yc(k) is the calculated response at observation point k. This form of error function is used because a zero value for E requires that the error be zero at every observation point (no cancellation of terms).

The minimization of the error may be accomplished by the methods of automatic optimization. A number of procedures are available, but the most common one is the method of steepest descent. Experience indicates that the method works nicely for class-1 problems. That is, it is not difficult
to find some set of parameters that gives a small error between theory and experiment. Class-2 problems are more difficult. Here the problem is one of uniqueness. The method is required to find the unique set of parameters that are the true “physical” parameters.

The steepest-descent method is essentially a systematic search in the multidimensional space in which the error function is related to the system parameters. The change in the error due to a change in parameters may be expressed using a Taylor series:

M

E = E0 + (dE/dXj)Ax,- + ■ ■ ■ (6.2.3)

i= 1

If the higher-order terms are ignored, then the relation between the error function and the system parameters is given by

M

E — E0 = £ (дЕ/дХі)Ax,- (6.2.4)

i= 1

This is valid for some region around the point where E0 and dE/dxt are evaluated. The greatest reduction in the error function for a given change in the parameters is obtained by making the parameter changes according to

Ax,-a ( — dE/dxi) (6.2.5)

That is, each parameter should be changed in proportion to the negative of the sensitivity of the error to that parameter. In the steepest-descent optimiza­tion, this gives a direction of change of the parameters that will reduce the error. This vector is explored by calculating the error at selected points along the vector. When the best point is found, a new direction is determined, and the process is repeated.

The sensitivities are found as follows:

se^_2y І уд) — ym ism

dxt Ye(k) ) dxi,

Of course, the experimental response does not depend on the assumed values for the parameters x,. The evaluation of the error sensitivity by Eq. (6.2.6) depends on the evaluation of the response sensitivity dYc(le)/<3x,. This may be done using the theoretical model and generalized techniques for evaluating sensitivity functions (54).

Criteria for uniqueness have been developed for the parameter identifica­tion problem. Buckner (40) studied this problem and considered the unique­ness of the identification of the coefficients in a linear state-variable model (a model consisting of coupled, first-order, linear differential equations). Sufficient criteria were derived for identifying some of the coefficients in a
system with n equations. It is necessary for all of the remaining coefficients to be correct. For the case of a measurement involving only a single input and a single output, it was found that:

1. Any (2n — 1) coefficient can be determined uniquely if the initial guesses on these adjustable coefficients are close to the correct values.

2. Any two diagonal coefficients and (n — 1) off-diagonal coefficients from any single row or column can be determined uniquely, regardless of the error in the initial guesses on these adjustable coefficients.

For the case of a test in which m different outputs that result from a single input are analyzed to give m separate frequency responses, it was found that:

1. Any m(n — 1) + n coefficients can be determined uniquely if the initial guesses are close to the correct values.

2. Any two diagonal coefficients and (n — 1) off-diagonal coefficients from any single row or column can be determined uniquely, regardless of the error in the initial guesses on the adjustable coefficients. This is the same result as for the test that uses a single output.

Other authors (28, 33) have studied the uniqueness problem for identifying the coefficients in a single nth-order differential equation. In this case, there are n coefficients to identify instead of up to n2 in a state-variable model. The general form of the equation is

d”x dn~1x d2x dx

~ПГ T Я„_ і. _ .—I" • • • + Й? ТТ T :—————- H QnX = f

dt" " 1 dtn 1 2dt2 1 dt

Buckner’s uniqueness criterion may be used if we convert this equation into an equivalent state-variable form:

dx/dt = Ax + /

where

0	1	0	0 ••	• 0
0	0	1	0 ••	• 0
0	0	0	1 ••	• 0
-flo	-fll	~a2	— a3 ••	‘ ~an-1

Since all of the adjustable coefficients are on a single row, all of them can be determined uniquely in a single test.

Another problem is the question of local optima versus global optima. Regardless of satisfaction of uniqueness criteria, the optimization procedure might find a local minimum rather than the absolute minimum. This is illustrated in Fig. 6.5 for a one-parameter problem. In general, there is no absolute test for global optimality. The only recourse is to repeat the search from several different starting points. If all the searches converge to the same optimum, then confidence is gained in the global optimality of the results.

Fig. 6.5. Sketch of error function versus a system coefficient.

Simple examples can illustrate the parameter estimation procedure (40). A model with known parameters was used to calculate the frequency response. This was used as the “experimental” response. The parameters were then changed and were used for a reference model. Then a steepest-descent parameter identification procedure was used to adjust the parameters in the reference model to improve agreement between “experimental” results and results from the reference model. The model used to give the “experi­mental” response was

dzl/dt = —z1 — 2 z2 + f dz2/dt = 0.4 z1 — 0.7 z2

The initial reference model for case 1 was

dzjdt = — 1.25zj — 2 z2 + /, dz2/dt = 0.65z! — 0.5z2

Figure 6.6 shows the “experimental” frequency response for bzjbf and the frequency response obtained from the initial reference model. In the parameter estimation calculation for case 1, the coefficient of z2 in the first

equation was not varied. The other three coefficients were adjusted to minimize the error. According to the uniqueness criteria, unique results should be obtained if the optimization converges to the global minimum. Figure 6.7 shows the current model parameters at each stage in the optimiza­tion. We observe that the final parameter estimates agree with the known true values.

Another example can demonstrate the uniqueness problem. In case 2, the model used for calculating the “experimental” model was the same as before. The initial reference model for case 2 was

dzx/dt = — 1.03zj — 1.8z2 + /, dz2/dt = 0.45z! — 0.7 z2

The coefficient of z2 in the second equation was held constant and the other coefficients were adjusted in the optimization. This indicates that the uniqueness criteria were not satisfied. Figure 6.8 shows frequency response results for SzJSf for the two cases. Figure 6.9 shows the current parameter

.001 .01 .1 1 10 100

(a) (b) Frequency (radians/sec.)

estimates at each stage in the optimization. Clearly, the estimated values are incorrect. However, the model based on these parameters give results that agree very well with theoretical results, as shown in Fig. 6.10. This type of problem with uniqueness suggests a cautious approach to practical parameter identification, but it does not mean that parameter identification cannot be very helpful if used properly.

Several workers have used parameter estimation techniques to determine important parameters in reactor systems. Cummins (36, 41) included this type of analysis in his work on the Dragon reactor. He used an optimization procedure to find the minimum squared difference between a theoretical step response and a measured step response. The minimum was found by a systematic adjustment of selected parameters in the theoretical model.

Obeid and Lapsley (42) determined the reactivity coefficients and heat — transfer coefficients in a swimming-pool reactor at the University of

Frequency (radians/sec.) Fig. 6.10. Amplitude for case 2.

Virginia. They obtained the zero-power frequency response and the closed — loop frequency response at power (0.9 MW). They then obtained the feedback frequency response from Eq. (6.2.2). In their fitting of the theoretical model to the experimental results, they took advantage of the fact that the three major temperature feedbacks (fuel, coolant, reflector) had significantly different relative importances in different frequency ranges.