Lecture
2.1. Setting Up the Experiment
Based on the testing method, identification methods are divided into active and passive.
A passive experiment is one in which the input and output data are recorded during normal operation without any additional intervention. It is used when the structure of the model is well known and its adequacy is not in doubt (i.e., when solving parametric identification problems).
When passive methods are used, the object operates under normal functioning conditions, and the model parameters are found from the results of statistical processing of the observations. The advantage of this approach is that there is no need to conduct special studies of the object — it is enough to measure the observed signals during the object's normal operating mode and then calculate the model parameters. The drawbacks of this approach are the considerable time required for data collection and the necessary statistical processing, as well as strict requirements on the frequency spectrum of the input action — it must not be lower than the frequencies of the dynamic characteristic of the object being identified.
An active experiment involves a special program of observations that, based on the results of the study, additionally allow the model structure to be estimated [5, 7]. In active methods, specially generated inputs — test signals, either deterministic or random in nature — are applied to the input of the object. The advantages of this approach are the minimal requirements on prior information about the object and the purposeful nature of the identifi-
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cation, which, as a result, reduces the time and material costs of conducting the experiment.
The factors of an active experiment are the variables that can be controlled and that are involved in building the model (xi).
Each factor can take on various values, called levels. In practice, the number of levels is infinite, or a continuous range of levels xi [x0i, xni].
In the theory of active experiments, this range is discretized and individual levels are selected [xi0, xi1, xi2].
A fixed set of levels is called a factor state. A design (plan) is a program for conducting the experiment that allows all factors to be used at all levels. If the design contains all possible combinations of factors and levels, it is called a full factorial design. If p is the total number of levels and k is the number of factors, then a full experimental design will include the following number of experiments:
experiments:
N = p^k.
(2.1)
Example: k = 3, p = 4, then N = 4^3 = 64 experiments.
A full factorial design allows an adequate model to be built, but it requires a large number of experiments; therefore, in practice, reduced designs are used — so-called fractional factorial designs — in which the number of experiments is smaller than in a full design, but they can still determine the adequacy of the model with a sufficient degree of accuracy.
Any model is determined by the formula
A = Y U
T
T
−1
(2.2)
U U
,
where M = U U^T is called the information matrix.
Depending on the method of minimizing the information matrix, the following designs are distinguished:
• D-design. An experimental design in which the choice of the information matrix is determined by the principle of minimizing the determinant —
min det Mi.
i
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• A-design. An experimental design in which the trace of the matrix is minimized —
min tr Mi.
i
• The condition for defining an E-design is choosing the design such that the maximum eigenvalue of the matrix M is minimal —
min max λ(M).
i
In practice, D-designs are most often constructed.
The main condition of a D-design is that it will satisfy the optimality condition if the information matrix M is diagonal.
−1
Example: p = 2, k = 3, xi = +1.
Accordingly, the full design will take the form
2^3 = 8
x1
x2
x3
–1
–1
–1
–1
–1
+1
–1
+1
–1
–1
+1
+1
+1
–1
–1
+1
–1
+1
+1
+1
–1
+1
+1
+1
Then, taking this condition into account, a certain fractional design is chosen: 2^(3–1) = 4
x1
x2
x3
–1
–1
+1
–1
+1
–1
+1
–1
–1
+1
+1
+1
i.e., we discard any one variable (x3), and the design is reduced by 4 units. For the remaining two variables a full design is constructed, and x3 is taken equal to x3 = x1 x2.
A D-design constructed on this principle satisfies the conditions of D-optimality, and the variable x3 is called the generator of the fractional design.
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For a two-level system, the number of experiments equals 2^(k–N), where N < k, and the remaining design generators are formed as the element-wise products of the basic factors, with the number of factors in each product ranging from 2 to N–k. Example: 2^(6–4)
x1
x2
x3
x1 x3
x2 x3
x1 x2 x3
–1
–1
+1
–1
–1
+1
–1
+1
–1
+1
–1
+1
+1
–1
–1
–1
+1
+1
+1
+1
+1
+1
+1
+1
Choosing two levels from a continuous range of levels:
xi [x0i, xni],
–1 – the start of the levels x0i; +1 – the end of the levels xni.
It should be noted that, in practical identification problems — in particular, in constructing identification models for subsystems of an aircraft engine control system (ACS) — the ability to vary the input variables (which is exactly what an active experiment represents) is severely limited, mainly by the design features of the object. For this reason, identification of ACS aircraft-engine models is carried out on the basis of a passive experiment.
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