Lecture
The reasonably well-developed theory of control system identification offers a large number of different research methods. However, none of the identification methods is universal — each is characterized by its own domain of applicability. The domain of applicability of a given method is largely determined by the amount of a priori information available about the object, which may include information on the object's structure, its noise characteristics, assumptions about the object's linearity and stationarity, and other data. Such information helps to compile a generalized characterization of the object under study, on the basis of which the identification methods appropriate to that class of object are chosen.
Formalizing the object of study (the hydromechanical unit of the aircraft-engine control system) reveals the following features:
•the structure of the object is known a priori and is determined by the design of the unit;
•the observed signals are continuous functions;
•the model is deterministic, since the level of measurement noise is small compared to the level of the useful signal. In addition, the information-measurement system performs additional filtering of the measurements;
•the model parameters are functions of the state variables, i.e., the model is nonlinear with respect to its parameters;
•the object is characterized by unmeasured coordinates;
•for the class of hydromechanical units under study, when identifying links it is necessary and sufficient to determine the time constants of the links and the overall (total) gain coefficient [26, 36, 37].
These features define the object as structurally determined but parametrically uncertain, deterministic, and nonlinear.
The simplest and most commonly used method in the design and identification of nonlinear multivariable objects is the two-stage identification method. In the first stage, the dynamics equations are linearized in the neighborhood of some base solution, and the problem of estimating the parameters of the linear system is solved. In the second stage, the models at the linearization points are approximated by some function.
The main advantage of this approach lies in its simplicity of implementation. Moreover, this approach makes it possible to directly account for the structure of the object, i.e., to obtain a model that is structurally similar to the object itself.
Since the problem of identifying the hydromechanical unit is solved within the framework of the inverse design problem, structural similarity between the object and the model is fundamental. Therefore, when identifying the hydromechanical unit, the following approach is preferable: combining linear algorithms with the patching (matching) method [1, 7, 15], which makes it possible to solve the identification problem for nonlinear systems of the class under study in two stages: identification of linear systems at the linearization points; approximation of the nonlinear coefficients. However, the presence of unmeasured coordinates in the object defines the identification problem as a joint estimation of parameters and state.
The presence of unmeasurable coordinates in the system requires checking the model for observability and identifiability.
3.3.1. Observability and identifiability of the pump-regulator subsystems
According to the observability criterion, a system is observable if
zank(Mн) = n,
(3.24)
where Mн is the observability matrix, Mн =
T
; С
T T
T
(А
n−1
T
( A –
С
А ;...С
)
coefficient matrix; С is the output matrix; n is the order of the system). According to the identifiability criterion, a system is identifi-
able if
zank(Mид) = n,
(3.25)
where Mид
is the identifiability matrix,
Mид = V(0); AnV(0);
... AnnV(0)
; V (0) is the initial value of the system's state vector, coeffi-
cients.
As shown above, the objects of study for the hydromechanical unit are limited to five types of models (3.19)–(3.23). For these links, the coefficient and output matrices have the corresponding form:
151
1. First-order lag link:
0
0
A =
,
b1
a11
C =[0
1].
2. Ideal integrating link:
0 0
A = b1 ,0
C =[0 1].
3. Real (non-ideal) integrating link:
0
0
0
a11
A = b1
0 ,
0
a21
0
C =[0 1 0].
4. Real integrating link with a summed input:
0
0
0
a11
A = b1
0 ,
a21
b2
0
C =[0 1 0].
5. Third-order link: series connection of a first-order lag link and a real integrating link:
0
0
0
0
a11
0
A =
b1
0
,
0
a21
a22
0
0
0
a32
0
C =[0
1 0
0].
152
Application of the observability and identifiability criteria [25] has shown that all subsystems of the pump-regulator are observable and identifiable.
3.3.2. Estimation of the pump-regulator model parameters
Estimation of the nonlinear model parameters is carried out in two stages:
•identification of the model parameters at the linearization points;
•approximation of the nonlinear coefficients.
3.3.2.1. Identification of model parameters at the linearization points
As shown earlier, the pump-regulator subsystems are described by 5 types of models. The first two types of links (first-order lag, ideal integrating) are first-order links that are fully measurable, so estimating the parameters of such links is not difficult and can be carried out using any algorithm, for example, the sequential regression method. The remaining links have unmeasured coordinates: for the 2nd-order link, one internal coordinate x, and for the 3rd-order link, two internal coordinates x1, x2.
It was shown earlier that existing identification methods do not ensure structural similarity between the model and the object, so the structure of the object must be taken into account when developing the algorithm.
Let us consider, for the link types under study, the transition matrices that take the structure of the object into account:
1. Real integrating link:
1
0
0
a11
Ф = b1
0 .
0
a21
1
2. Real integrating link with a summed input:
1
0
0
a11
Ф = b1
0 .
a21
b2
1
153
3. 3rd-order link: series connection of a first-order lag link and a real integrating link:
1
0
0
0
a11
0
Ф =
b1
0
0
a21
a22
.
0
0
0
a32
1
In accordance with the above algorithm, for 2nd-order links the equation describing the system dynamics has the form
с1 y(k +2) +с2 y(k +1) +с3 y(k) = u(k),
(3.26)
where the coefficients сi for a real integrating link are as follows:
b1a21 = 1 ,
c1
a11 +a22 = − c2 ,
c1
a11a22 = c3 ,
c1
for a real integrating link with a summed input:
b1a21 +b2 (1−a11) = 1 ,
c1
a11 +a22 = − c2 ,
c1
a11a22 = c3.
c1
In accordance with the above algorithm, for the 3rd-order link the equation describing the system dynamics can be written as
с1 y(k +3) +с2 y(k +2) +с3 y(k +
1) +с4 y(k) = u(k),
(3.27)
where the coefficients сi have the following form:
b a
21
a =
1
,
1
32
c1
154
a11 +a22 +1 = − c2 ,
c1
a11a22 +a11 + a22 = c3 ,
c1
a11a22 = − c4 .
c1
In the study of the subsystem under consideration — the speed governor — the proposed algorithm can be applied to the isodrome regulator, which is a real integrating link with a summed input.
The mathematical model of the isodrome regulator has the form dxdt4 = b1u +a11x4 ,
dxdt5 = b2u + a21x5 ,
where x4 is the displacement of the isodrome; x5 is the displacement of the DI (differentiating element);
u is the reduced displacement of the sensor rod and the slider. The transition matrix of the isodrome regulator has the form
1
0
0
a11
Ф = b1
0 .
a21
b2
1
The range of variation of the isodrome displacement x4 is from 0.3 to 3 cm.
For identification purposes, the following linearization points of the static characteristics have been defined:
Point No.
х40
u0
P30
of linearization
1
0.3
0.0900
10.474
2
0.6
0.1858
10.947
3
0.9
0.2629
11.421
4
1.2
0.3487
11.895
5
1.5
0.4257
12.368
6
1.8
0.505
12.842
7
2.1
0.7347
13.316
8
2.4
0.8143
13.789
9
2.7
0.8920
14.263
10
3.0
0.9682
14.737
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At each linearization point, a transient process is considered for a small increment u = u0 +∆u.
At each point, the system parameters were estimated using the algorithm described above.
The estimation results are presented in the table:
Point No.
b1
b2
a11
a21
of linearization
1
42.160
12.684
–13.879
–1.771
2
42.300
12.801
–13.270
–1.781
3
42.504
12.923
–12.936
–1.717
4
42.618
13.001
–21.034
–1.701
5
42.795
13.089
–20.475
–1.702
6
42.894
13.102
–19.216
–1.715
7
43.008
13.186
–17.841
–1.5598
8
43.088
13.216
–17.110
–1.625
9
43.126
13.350
–16.576
–1.666
10
43.227
13.441
–16.197
–1.722
3.3.2.2. Approximation of the nonlinear coefficients
Analysis of the nonlinearities characteristic of the hydromechanical unit, from the standpoint of parameter estimation, makes it possible to divide them into two groups [34, 35]:
•monotonic nonlinearities (centrifugal speed sensors, elements with variable profiles, etc.);
•nonlinearities with first-kind discontinuities (connection of additional throttle packs, variable limits, etc.).
Approximating monotonic nonlinearities presents no particular difficulties.
When approximating nonlinearities with first-kind discontinuities (Fig. 3.4), methods of approximation by monotonic functions can be applied over the entire domain of study except at the discontinuity point. The distinctive feature of approximating a piecewise-continuous function lies
in determining the approximating functions fij1(u) and fij2(u) in the contin-
uous regions, determining the discontinuity point, and determining the value of the approximating function at that point according to the principle: if the value of
the argument u is increasing, then fij(uразр) = fij1(uразр); if the value of the argument u is decreasing, then fij(uразр) = fij2(uразр).
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Fig. 3.4. Graph of a piecewise-continuous dependence with a first-kind discontinuity
In addition to monotonicity, the nonlinear characteristics of the hydromechanical unit may depend on different numbers of arguments.
Depending on the number of variables, the following nonlinearities can be distinguished:
1. Dependence on one variable:
• fij = f0 + f1u.
This type of nonlinearity is characteristic of the coefficients of the isodrome regulator model, the physical speed sensor, and the starting automatic device.
• fij = f0++ f1u. 1 f2u
This type of nonlinearity is characteristic of the coefficients of the model of the constant-pressure-drop valve, the fuel-flow control unit operating on signals from the main automatics, and the IGV (inlet guide vane) control unit operating on signals from the main automatics.
• fij = f0 +1f+1uf+3uf2u2 .
This type of nonlinearity is characteristic of the coefficients of the model of the constant-pressure-drop valve, the fuel-flow control unit operating on signals from the main automatics, and the IGV control unit operating on signals from the main automatics.
2. Dependence on two variables:
• fij = f0 + f1u1 + f2u2.
This type of nonlinearity is characteristic of the coefficients of the model of the IGV control mechanism on the backup automatics and on signals from the main automatics.
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When approximating the nonlinear coefficients of the isodrome regulator model, the following should be borne in mind.
The transition matrix of the isodrome regulator model has the form
1
0
0
a11
Ф = b1
0 ,
a21
b2
1
where b1 = b1(P3), b2 = b2(P3),
a11 = a0(x4) + a1(P3)u, a21 = a2(x4) + a4(P3)u,
u is the reduced displacement of the sensor rod and the slider; P3 is the operating pressure.
The numerical value of a0 is determined by the discharge coefficient Qиз, which includes a parallel connection of two throttle packs: the main one Q1 and the additional one Q2. Connecting the additional throttle pack Q2 when the isodrome piston moves causes a stepwise increase in Qиз, and consequently the dependence a0(x4) is characterized, at the moment the additional throttle pack is connected, by a discontinuity point of the first kind (Fig. 3.5). The dependence a11(P3, u) is likewise characterized by the presence of a discontinuity point (Fig. 3.6).
Fig. 3.5. Graph of the nonlinear dependence a0 = f(x4)
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Fig. 3.6. Graph of the nonlinear dependence a11 = f(u)
The results of approximating the nonlinear coefficients of the isodrome regulator model are given below:
Characteristic
b1
b2
a11
a21
a0
a1
a2
a3
Before the discontinuity point
43.008
13.102
–13.564
3.007
–1.965
0.9015
After the discontinuity point
43.008
13.102
–19.841
2.737
–2.213
0.804
An analysis of the adequacy of the identification models relative to the original object showed that the model matches the object with an accuracy of no worse than 1.4%. The presence of discontinuous characteristics in the object increases the error in estimating the model parameters, with the maximum error observed at the discontinuity points, but it does not exceed 1.6% across all operating modes examined for the identification models.
Thus, the described approach to constructing identification models makes it possible to develop models that are adequate to the original objects with an accuracy of no worse than 1.6%, which meets the engineering requirements imposed on the solution of design problems.
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