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SOLUTION OF THE PROBLEM OF OPTIMAL SYNTHESIS OF A POSITIONING CONTROL SYSTEM FOR A STOCHASTIC MULTIDIMENSIONAL AIRCRAFT MOTION SIMULATION PLATFORM
02.10.2025 14:02
[3. Technical sciences]
Author: Oleksandr Sieliukov, Grand Dr. in Engineering, Professor, School of Aerospace Engineering, Xi'an Jiaotong University, Xi'an, China, State Key Laboratory for Strength and Vibration of Mechanical Structures; Enbo Yang, Ph.D. student, School of Aerospace Engineering, Xi'an Jiaotong University, Xi'an, China, State Key Laboratory for Strength and Vibration of Mechanical Structures
ORCID: 0000-0001-7979-3434 Oleksandr Sieliukov
ORCID: 0009-0007-6655-5796 Enbo Yang
Systems for simulating the natural flight of aircraft are widely used for various purposes. These include certifying aviation system sensors, configuring and testing autopilots and onboard control systems, and researching human psychophysiological characteristics, among others. The relevance of developing and creating such systems is underscored by the large number of companies engaged in developing similar systems. Virtually all design bureaus in the aerospace industry have full-scale simulation systems in their structure. Despite this, the accuracy of natural modeling of stochastic flight conditions is limited. This limitation leads to increased time, labor, and cost of developing onboard equipment. The primary reason for these limitations is the imperfection of the platform motion control system, which simulates the behavior of the simulation object.
Most natural flight simulation systems for aircraft consist of a computer and a platform (test bench) for its movements. They are connected via a special interface. Almost all of them are designed using an open-loop structure (Fig. 1).
Figure 1. Standard design of an aircraft flight simulator
As is known from various sources, for example [1], at each stage of flight, the aircraft oscillates around a given trajectory. These oscillations usually belong to a set of multidimensional stationary random processes. The dynamics of these processes differ significantly from white noise. In addition, the identification of the dynamics of perturbations (ψ) and noise (φ) showed [1-3] that they also belong to a similar set of random processes.
In this regard, the use of time methods requires an increase in the number of state variables while keeping the size of the control signal vector unchanged. As a result, the effect of loss of observability and controllability of the installation may occur. The use of existing frequency methods, in turn, is limited to control systems that do not have a transfer function matrix. Consequently, a new task arises: synthesizing an optimal multidimensional control system.
Let the dynamics of the platform be described by a system of ordinary differential equations with constant coefficients of the form
where P0 is a polynomial matrix of dimension n0×n0 from the differentiation operator p;
x0 is the vector of coordinates defining the stand platform position, which has dimension n0;
M0 is a polynomial matrix of dimension n0×m0 from the differentiation operator p;
u0 is the stand drives control signals vector, which has dimension m0;
ψ0 is the disturbance vector.
Let us also assume that the inverse kinematics problem is solved in the system with sufficient accuracy; therefore, the matrix of transfer functions IC is equal to unity, and the interpolation noise φ is sufficiently small. In this case, the structural diagram of the platform control system (Fig. 1) can be represented as follows (Fig. 2).
Figure 2. Asymplified closed-loop structure
The probing signals vector r is a vector of uncorrelated unit-intensity white noises. The disturbance vector ψ0 is a centered stationary random process with a known transposed matrix of spectral densities SψψТ. In this case, the synthesis problem is to find, for given matrices FF, M0, P0 and SψψТ, such controller transfer function matrices W1, W2, W3, which ensure the following two conditions:
- the closed-loop system (Fig. 2) is stable;
- the following quality criterion reaches a minimum
where e – a modeling errors vector
R0 – a positively defined weight matrix;
u – a control signal vector
С0 – a non-negatively defined weight matrix;
<> - a mean symbol;
Т - a transposition sign.
For the structural diagram shown in Fig. 2, a system of equations can be formulated. It connects the Fourier images of the input and output signal vectors of the system at zero initial conditions and has the form
The transfer function matrix FF can be represented as the product of two polynomial matrices
using left-side pole removal [4] or MFD decomposition [5], allowed one to rewrite the equations system (5) as follows
In order to transform the system of equations (7) to a vector-matrix form, the following notations were introduced
where En0 is the identity matrix of dimension n0×n0;
On0 is the zero matrix of dimension n0×n0;
On0×m0 is the zero matrix of dimension n0×m0.
Taking into account the notations (8), (9) allows one to write the control object equations system in a vector-matrix form:
The controller, consisting of three parts transfer functions matrix, taking into account Fig. 2, is represented as follows
Taking into account the designations (8), equation (10) and block matrix (11), the structural diagram (Fig. 2) is presented in the form of a typical multidimensional stabilization system diagram (Fig. 3), and the synthesizing the optimal control law for the aircraft movement simulator platform position problem is reduced to the synthesizing a multidimensional stochastic stabilization system task.
Figure 3. A symplified closed-loop sructure
Such a system optimality criterion (2), taking into account the accepted notations, written according to [4] in the frequency domain, can be represented as follows:
where Fx is the matrix of transfer functions from the vector ψ to the vector x;
Fu is the matrix of transfer functions from the vector ψ to the vector u;
tr is the trace of the matrix;
j is a complex unit;
index * is the Hermitian conjugation sign;
SψψT is the vector ψ spectral densities transposed matrix;
s is the complex variable s=jω;
R is a 2n0+m0×2n0+m0 dimension weight matrix of the form
In this case, finding the optimal controller transfer functions matrix W can be done with one of the algorithms described in the sources [4, 6].
According to the algorithm in the article [6], the matrix W that meets the synthesis problem conditions can be found from the equation:
where A, B are auxiliary matrices, that are found as a result of representing the block matrix H as a two-factor (block matrices) V and Σ product,
The matrix H on the left side of equation (16) is equal to
The block matrix Σ is numeric and has the form
The polynomial matrix V, which is found as a result of applying the algorithm from [7] to the matrix H, consists of the following blocks
The matrix Ф from equation (15) is a fractional rational stable transfer function matrix. Its search was carried out as a result of using the Wiener-Kolmogorov method to minimize the functional (12). The result of the minimization is the following rule for calculating the matrix Ф
where T0+T+ is a stable result of separation (splitting) [4] of the matrix product:
D is the factorization [4] result of the spectral densities transposed matrix equivalent external influences vector ψ
If the optimal controller transfer functions matrix W and the structural diagram (Fig. 2) are known, the closed loop system transfer functions matrices Fx and Fu can be found, for example:
These matrices make possible analyzing the aircraft motion modeling quality.
The presented material proves the possibility of significantly increasing the aircraft flight dynamics modeling accuracy using a simulator platform in laboratory conditions. To achieve this goal, it is proposed to use a feedback control system, which regulator should consist of three parts.
The search for such a controller transfer functions, ensuring stability and modeling high quality, can be carried out by using methods for synthesizing the rigid body motion stabilization optimal multidimensional systems. To ensure such an opportunity, an original method of structural transformations has been developed and presented.
Further research should be directed towards the development and implementation of multidimensional optimal robust control systems for solving simulating the movements of moving objects under stochastic conditions problems.
REFERENCES
1. Thomas R. Yechout. Introduction to Aircraft Flight Mechanics: Performance, Static Stability, Dynamic Stability, and Classical Feedback Control. AIAA Education Series. Editor-in-Chief Joseph A. Schetz. American Institute of Aeronautics and Astronautics, Inc.1801 Alexander Bell Drive, Reston. VA 20191-4344 – 631 p.
2. Flying Qualities of Piloted Aircraft. Department of Defense Handbook. MIL-HDBK-1797B. Washington. DC: U.S. Department of Defense. 2012.
3. Hoblit, Frederic M., Gust Loads on Aircraft: Concepts and Applications. Reston, VA: AIAA Education Series. 1988.
4. F.A. Aliev, V.B. Larin. Optimization of Linear Control Systems. Analytical Methods and Computational Algorithms. Overseas Publishers Association. N.V. 1998.
5. Kucera V. Discrete line control: the polynomial equation approach. – Praha: Akademia, 1979. 206 p.
6. Osadchiy S.I. Combined method for the synthesis of optimal stabilization systems of multidimensional moving objects under stationary random impacts / S.I. Osadchiy, V.A. Zozulya // Journal of Automation and Information Sciences. Vol. 45. 2013. Issue 6. pp. 25-35.
7. Науменко К.И. Наблюдение и управление движением динамических систем - Киев: Наукова думка. 1984. 208
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