2.1. Classes of Mathematical Models of Continuous–Discrete Systems
If discrete subsystems operate on the same time sequence, then such a continuous–discrete automatic control system is called single-rate [
16]. In [
17,
18,
19,
20], such a linear automatic control system is described by a set of differential equations describing the control system on intervals of continuity
, and difference equations describing state jumps at discrete times
:
where
are the matrices of the coefficients of the continuous part of the system,
is the continuous control vector,
are the matrices of the coefficients of the discrete part of the system,
is the vector of the discrete control,
is the
-dimensional vector of the state of the stabilization system, the elements of which are continuous functions on finite time intervals
with discontinuities of the first kind at times
,
—the value of the state vector at the moment of action of the pulse quantizer:
—the set of times at which the state vector undergoes discontinuities.
The solution to systems of Equation (1) is piecewise continuous functions that satisfy the differential equations of the system at finite time intervals , and at times from the set undergo discontinuities of the first kind following the given finite difference equations.
In many problems, discrete subsystems of continuous–discrete UAV stabilization systems operate on different time sequences. This is due to both the functional features of discrete subsystems and the need to achieve their maximum efficiency. Therefore, for example, in an onboard computer of an aircraft-type UAV, signals from angular velocity sensors are processed much more often than signals about the position of the center of mass.
It is shown in [
16] that the general model described by the system (1) allows one to describe the dynamics of multitasking automatic control systems since the time sequence in it is arbitrary. To explicitly highlight the subsystems and take into account the peculiarities of the dynamics of the UAV stabilization system, it is advisable to introduce a mathematical model of multi-rate continuous–discrete control systems.
The general model of a multi-rate continuous–discrete stabilization system in the time domain can be represented as
In system (3), the following designations are accepted:
—the state vector of the continuous part of the control system;
—the state vectors of the discrete subsystems of the stabilization system;
—the matrices of the coefficients of the continuous part of the control system;
—the matrices of the coefficients of the discrete subsystems of the automatic control system;
—the matrix of the coefficients of the discrete part of the system affecting the continuous;
—the current time functions:
—a set of moments in time that determine the functioning of discrete subsystems of UAV stabilization systems, where:
and
is the sampling period of the
-th discrete subsystem of the UAV stabilization system.
The simplest case is when the sampling periods of discrete subsystems (6) are mutually rational numbers, which is most often encountered in practice. For example, if the discrete subsystems of the stabilization system are implemented in the form of programs of one control computer, then their discreteness cycles contain an integer number of sampling periods of the computer’s clock generator, and therefore, are mutually rationally simple.
A great development in the issue of constructing mathematical models of multiloop but not multidimensional control systems with different sampling periods specified using transfer functions (based on the z-transform and discrete Laplace transform), despite the great complexity of the proposed approaches and serious restrictions on their application, was achieved thanks to works [
13,
21,
22]. This approach is very relevant due to the ability to work with models in the form of structural diagrams.
In [
23,
24], a mathematical model of a multiloop multidimensional system is obtained in the operator vector–matrix form, which takes into account the influence of all quantizers of the system. This approach is fully applicable to the description of the UAV stabilization system, which, as mentioned earlier, is a multiloop multidimensional continuous–discrete system. For completeness, we present the construction of these mathematical models [
23,
24].
Let be the vector of control actions on the object of dimension, and —the vector of outputs of the object of dimension. Let be a vector of variables quantized on an analog-to-digital converter with sampling periods , respectively (among which there may be equal ones). We will consider commensurate sampling periods that are multiples of a certain sampling period , i.e., where are natural numbers.
The equations of a continuous object and analog circuits of the stabilization system from the object to the quantization keys have the form:
where
are the matrices of transfer functions of the corresponding dimensions. Thus:
Let
—discrete Laplace transforms quantized concerning the sampling periods
of the variables
, respectively. Each of the quantized signals
is converted by a corresponding digital circuit and summed with other similar signals, as a result of which a control action is formed. In addition to digital circuits, analog circuits (from the outputs of the object) can also be used in the formation of control actions. Let us write the equation for the “
k-th” component of the vector of control actions:
where
are the transfer functions of parallel digital and analog circuits, and
is the reference action generated by the digital circuit.
With the digital summation of converted digital signals (with sampling period
and corresponding signal repetition) and subsequent digital-to-analog conversion, it can be assumed that:
where
is the periodic (with a period
) transfer function of the digital (impulse) conversion, and:
where
is the Laplace transform of the quantized reference action–periodic (with a period
) function.
In a more general case of combined digital and analog (using other signals) summation, it is also possible to accept relation (10) to describe digital feedback circuits with a corresponding complication of the periodic part of the transfer function, for example, by writing:
where
—periodic part of the transfer function
.
Let us introduce a matrix with elements and vectors with elements and . The symbol “*” at the bottom marks the property of periodicity of the matrix and the vector for the corresponding periods. At the same time, all these elements satisfy relations of the form .
Taking into account relations (10) and (11) as rather general and writing down the equations for control actions in vector–matrix form:
in which
is a
matrix with elements
, we obtain, taking into account relations (7) and (8):
where
.
Relations (12), (13), (14) together with the vector represent a model of the closed multiloop, multidimensional multi-rate continuous–discrete UAV stabilization system, which takes into account both the influence of all quantizers of the system and its multiloop, including the cross-connections of control channels. Note, as indicated above, that the use of mathematical models of automatic control systems in the form of transfer functions allows using models in the form of structural diagrams.
2.3. Influence of Deformations of the UAV Body on Its Dynamics
The transfer functions of elastic links characterize the effect of the dynamics of stabilization of elastic vibrations of a moving UAV. This effect can be quite noticeable.
All mechanical rod structures, including the UAV body, are not rigid structures that vibrate [
26,
27]. The main reason that causes bending and flexural vibrations of the body is the control torque generated by the steering elements. This becomes especially important for pumped aircraft because such objects tend to be lighter in weight. To increase the UAV’s flight range, the most acceptable solution is to increase its length, which in turn increases the flexibility of the body. The elasticity of the UAV body structure harms the stability and quality of the control system.
A large number of studies have been devoted to the study of flying vehicles as elastic bodies, for example [
28].
In [
29], it is shown how oscillations affect the motion of an aircraft. For example, under the influence of a disturbing moment
, the aircraft is deflected by a certain yaw angle
. Then, the stabilization machine, turning the steering elements (
), will create a control torque
, which should compensate the harmful effect of the disturbance (see
Figure 2). However, the body of the aircraft, as already mentioned above, is not an absolutely rigid structure, and it will bend under the action of the torque
, as a result, the yaw angle sensor, placed on the gyro platform, will register the angle not the angle
, but
(see
Figure 3).
For the analysis of oscillations, the transfer function of the UAV is obtained as an object of regulation, taking into account the oscillators, which is not always convenient, since they are usually satisfied with approximate models of the phenomenon in the form of several vibrational links corresponding to the fundamental tones of elastic vibrations. The question of the accuracy of such approximations of the model is solved mainly by experimental means in each specific case. In this regard, there is a problem with constructing an accurate dynamic model of an elastic link under certain assumptions. This will make it possible to compare the exact and approximate characteristics of the elastic link and, therefore, reasonably introduce the corresponding simplifications.
The elastic properties of a UAV can be significantly manifested in its movement dynamics. Transient processes in the stabilization loop, occurring under the action of aerodynamic forces, are accompanied by the elastic deformations of the UAV body, which affect the signals of the measuring devices. For example, the bending angles of the body add an additional component to the signals of gyroscopes, and the accelerations of elastic vibrations are manifested in the signals of accelerometers. Due to this, additional feedbacks–elastic links appear in the stabilization loop, which should be taken into account when analyzing the properties of the loop. The mathematical description of these links, as well as the subsequent analysis, taking into account the stabilization loop, is a very difficult task. Usually, they are satisfied with approximate models of the phenomenon in the form of one or more vibrational links corresponding to the fundamental tones of elastic vibrations. Below, there is a description of the principle of constructing a mathematical model that describes an elastic link in the form of transfer functions.
Methods for constructing transfer functions are given in [
30,
31,
32]. Furthermore, when describing the construction of a mathematical model of the UAV stabilization system, taking into account elastic vibrations, we will briefly give a method for constructing transfer functions describing an elastic link.
A model of the stabilization system, the block diagram of which is shown in
Figure 1 will be called “the complete model”. Sampling periods are assumed to be commensurate, i.e., multiples of a certain number
—the greatest common divisor of sampling periods. Otherwise, the problem of analyzing the model becomes rather complicated, and as been mentioned earlier, practically insoluble. However, usually, due to the operation of one control computer onboard the UAV, the condition of the frequency of sampling periods is satisfied. “The complete model” is the basic model for obtaining a family of simplified models of the UAV stabilization system.
2.4. Hierarchical Models of the UAV Stabilization System
It is natural to develop methods for the synthesis of multiloop multi-rate continuous–discrete UAV stabilization systems according to the principle “from simple to complex.” The degree of complexity is determined by the degree of complexity of the model. Therefore, it is natural to introduce into consideration a certain hierarchy of models with varying degrees of simplification obtained from the complete model under various kinds of assumptions. Assumptions apply to those aspects of the overall model that determine the complexity of the synthesis and analysis problem.
The meromorphism of the transfer functions greatly complicates the task of synthesizing and analyzing the system. If elastic links are described approximately in the class of rational functions, a simplified model follows from the complete model, which we will call “model 1”.
Disregarding the effect of quantization in digital feedback loops of the full model results in “model 2”. By separating the angular motion stabilization system from it, we obtain a block diagram, which is shown in
Figure 4. “Model 2” is a continuous two-dimensional stabilization system taking into account the elastic links of the object. If in “model 2” we neglect the mutual influence of the heading and roll channels, we obtain a continuous one-dimensional model of the stabilization system. We will call it “model 3”. The block diagram of “model 3” is shown in
Figure 5.
Finally, neglecting the elastic constraints in “model 2”, we obtained “model 4”. Its structural diagram is identical to the structural diagram shown in
Figure 4, at
. Synthesis and analysis of complex multiloop UAV stabilization systems with specific signs of complexity are carried out following the given hierarchy of models.
Let us consider the application of the proposed approach, for example, for analyzing the stability of the UAV stabilization loop. By “model 3”, the stabilization circuit of a “rigid” unmanned aerial vehicle can be considered in the form shown in
Figure 6. In this figure,
ψ—the yaw angle of the vehicle;
—the rudder deflection angle;
—the specified yaw course angle; and
—the stabilization error.
The dynamics of stabilization are determined by the transfer functions of
—the control object (UAV in rotational motion around the center of mass); and
—the feedback loop of the stabilization system. When small elastic deformations occur that do not change the aerodynamic forces of the UAV occur, the transfer function
of the object remains the same, but the stabilization contour changes as new connections appear. If the sensitive element of the feedback circuit is a gyroscopic device that responds to the yaw angle, which characterizes the position of the vehicle body relative to the constant direction, then the modified contour will have the form shown in
Figure 7.
In this figure, is the transfer function of the elastic connection of the object, determined by the equality , —rudder angle.
To determine the transfer function, we note that, according to the equations of motion of a rigid apparatus, it is possible to establish the relationship in the images between the angle of sliding of the apparatus and the angle of deflection of the steering wheel:
where
is the corresponding transfer function of the “rigid” UAV.
In this case, we take into account that in the presence of a free gyroscope, an additional signal of this sensitive element arises, since it will register the bending angle of the apparatus body. For small vibrations, this angle can be defined as
where
is the abscissa of the gyroscope attachment point.
Differentiating by
the expression for the deflection, we find:
where
,
,
—sliding angle and:
represents the transfer function of the UAV as an elastic link.
Then, the ratio for the bending angle of the UAV body can be given the form:
Fixing in this expression the value at the given position of the gyroscopic device in the UAV
, and taking:
we obtain the relation for
, where
.
A more complex situation arises when the UAV stabilization system contains two gyroscopic devices located in different places—a free gyroscope and a gyrotachometer. In this case, each device reacts to the UAV bending angle in its place. The block diagram takes the form shown in
Figure 8.
Here, each of the transfer functions
is determined by an expression of the form (20) with the values of the abscissa
, respectively. If
coincide, the elastic links are combined, and:
Similarly, we can consider the structural diagrams of the stabilization system when used in the system of accelerometers.