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PRINCIPLES OF CREATION OF SOFTWARE FOR PROCESSING FLIGHT INFORMATION AT THE FLIGHT TEST STAGE OF AEROSPACE SYSTEMS
Genady Burov, Dr.sc.ing., Lead researcher
Riga Technical University
Information about dynamic characteristics of aircraft and their onboard equipment is difficult to access because of weak variability of measured transients, which are damped by special devices for purposes of aircraft flight safety. However, acquiring such information allows improving the program of flight tests, reducing their duration and expenses. Such information is also necessary to obtain during aircraft operation with the purpose of diagnosis of conditions of onboard equipment and duly indication of occurrence of abnormal situations in its functioning. Traditional algorithms are not adapted for working in such conditions because singular situations arise in them breaking their serviceability and leading to reception of unreliable results. Taking into account the feature of aircraft connected to the deficiency of dynamism of transients because of their damping, it is offered to use new algorithms for processing the flight information that are capable to work in real time of flight modes. The method for description of computing algorithms as symbolical combinatory models (SC-models) has been developed. They can be used for optimization of algorithms and development of formalized mathematical methods for creation of software for aircraft onboard computers, allowing to realize parallel modes of processing of results of measurements of signals in conditions of deficiency of their dynamism.
Keywords: Identification, technical object, aerospace object, symbolical combinatory model, parallel algorithm, graph structure.
The abovementioned features require rethinking of the problem of creating software for processing of flight information. For this purpose it is offered to use new forms of description of computing algorithms on the basis of which software should be created. At that, the ability of algorithms to give reliable estimates of identification in conditions of low variability of processed transients taken from sensors of aerospace object should be taken into account. Use of reliable software allows taking the on-ground digital processing systems to the level where some characteristics of aircraft and its equipment can be reliably determined even without flight tests. The problem reduces to sufficiently extensive determination of real conditions and initial parameters that should be simulated during modeling the flight modes for getting reliable results. Work in these directions can lead to reduced number of basic modes that should be included in the flight test schedule and, consequently, to reduced duration of the test stage and reduced expenses.
The main part of the software constitute the algorithms related to soving equation systems formed from results of signal measurements:
that are taken from the sensors of the aerospace object. They contain information about dynamic characteristics of the identified analog object which is expressed by coefficients of its transfer operator:
The aprioristic information about operator (2) is usually known already at the design stage of the object and it can further be used for computer-based diagnosis of its condition. Confirmation about coefficients of operator (1) should be found from information obtained in conditions of real flight modes. For this purpose, the system of equations generated from measurements (1) is solved:
using an algorithm related to the discrete generating operator:
It is the discrete equivalent of object’s differential equation and shows the discrete character of processing of analog information (1). Transition from (2) to (4) is found on the basis of mathematical operation of Z-transform in which an interpolating operator for compensating discrete errors of discrete approximation is included.
Solving (3) gives an estimate of vector of coefficients of characteristic polynomial (4):
They contain information about dynamic characteristics of the object, about its eigenfrequencies and attenuation decrements. However they act in an implicit form as abstract numbers which are not reflecting the physical condition of the object. It is visible from the character of nonlinear mathematical relation between poles of operators (1) and (3):
From it follows that there is a significant compression of information: analog poles of (2) located in the left infinite semi-plane and therefore well separated at the presence of noise are mapped into discrete poles located in a narrow area of the right unit semi-circle that are poorly separated at the presence of noise. With the increase of sampling frequency of signals (1), as it follows from (6), they all are pulled together into a single point and perceived as one multiple pole of operator (4) leading to algorithmic uncertainty and impossibility of reliable restoration of operator (2) from experimentally obtained estimation (4). Therefore, the statement that by increasing the sampling frequency of measurements of signals (1) it is possible to increase the accuracy of identification is wrong. For the same reason, introduction in stochastic models of identification  additional operators in a subjective way with the purpose of determining the characteristics of noise increases the algorithmic uncertainty even more. Besides that, there are also other proofs about impossibility of practical application of such models, as it is detailed in .
In software it is necessary to introduce operation of decoding of solution of system of difference equations (4) and operation of checking the reliability of its results by introducing information feedback loop in circuit of identification. On its basis, it is necessary to calculate the output signal from experimentally determined values of coefficients of operators (2) and (4) and estimate its difference from the experimental one recorded during flight mode. Recommendations to use discrepancy of difference equations  are methodically erroneous because they are based on the method of fitting them on the basis of coefficients that have no physical interpretation. From (3) and (5) follows that for realization of algorithm of decoding, it is necessary to apply nonlinear operation of inverse Z-transform:
However it is characterized by numerical instability and, consequently, a high sensitivity to noise. It does not guarantee getting authentic results of decoding. Therefore in -, alternative methods of decoding based on principles of mapping of mathematical objects in information spaces specified in the circuit have been offered. For their realization, the method of Fourier decomposition of experimentally obtained estimation of vector (5) in the basis generated from reference values can be used. Such approach opens ample opportunities for application of formalized mathematical methods in the software for using aprioristic information in processing the flight information.
The main principle of creation of software for decoding the flight information is avoiding usage of algorithms based on solving numerically unstable mathematical problems of inverse type, for example, in which sensitivity of the solution to influence of noise of various types is large (including rounding errors in the computing process). The factor of amplification of noise can vary in an unpredictable way depending on the character of experimental information. For example, if the matrix is generated in such “steady” flight mode where the signal (1) is almost constant, getting the necessary information about dynamic characteristics of aerospace object by solving the system (3) will hardly be possible. Such flight mode is safe, but useless. On the other hand, creating sharp evolutions of aircraft movement while not overstepping the bounds of flight restrictions is not always possible. It is necessary to solve a complicated problem in conditions related to the significant risks arising during carrying out flight tests.
More safely it can be solved by using the aprioristic information about the object collected during the previous period of flight tests and also during design and operation together with corresponding software related to identification algorithms. This approach was not commonly used because of mathematical formalization of the aprioristic information. However it can be realized on principles of monitoring between information spaces in the circuit of which the mapping of mathematical objects used at identification is carried out. For this purpose it is necessary to use spaces of experimental and calculated values of signals (1) and calculation spaces of coefficients of operators and . Existence of determined mathematical relations between these mathematical objects guarantees existence of isomorphism between information spaces in which they are contained. This property means that there is an isomorphism between Fourier systems containing spectral coefficients of decomposition of experimentally obtained objects in the bases obtained in the calculated way. For example, in the problem of decoding of the abstract vector (5) it is appropriate to use systems of equations generated in calculation Fourier bases of signals and vectors :
More in detail the method is stated in papers , . There a technique for checking the degree of isomorphism is given. For its increase and, accordingly, for increase of reliability of results of decoding the method of introduction of additional calculation spaces in procedure of monitoring is offered. In  efficiency of this method has been confirmed on the basis of imitating modeling of identification process of characteristics of an aircraft engine. Realization of the method is important from the point of view of acquiring the information about attributes of occurrence of compressor
From (8) it is visible that necessity of solving ill-conditioned inverse problems still remains at calculation of inverse matrices for Fourier bases. However there is a possibility to improve their conditionality as there is a freedom of choice of calculation bases. But for overcoming these computing difficulties the methods connected to application of numerically stable symbolical combinatory models for calculation of inverse matrices which is stated below can be used.
A particular feature of aerospace objects is that transients are usually damped by special devices for flight safety reasons. Therefore, for example, in airplanes for reducing angular fluctuations in all three channels of pitch, yaw and roll, dampers of angular fluctuations are used. The same applies also to booster rockets where elastic fluctuations of rocket’s body and fluctuations of liquid propellant in fuel tanks are damped. It means that for processing such damped signals it is necessary to invert ill-conditioned matrices (4) and it creates difficulties in getting reliable estimations of (4). Traditional computing algorithms are not suitable for their inversion as thus unpredictable singular situations can arise resulting in sharp increase of errors.
Matrices are Toeplitz matrices that worsens their conditionality even more. Therefore instead of traditional algorithms for finding the inverse of ill-conditioned matrices it is appropriate to apply alternative algorithm in which singular situations related to division of results obtained in separate stages by small values that are close to the level of noise do not arise, as it happens in Gaussian method. According to it, the elements of the inverse matrix are found as minors that are calculated using the known formula:
Here, the sum is distributed over all possible permutations P = (?1, ?2, …, ?n) of elements of Y: 1, 2, …, n. The elements of (5) are found as minors of from which the matrix H is formed:
Elements of H are found using the formula (10) in which permutations are formed with the help of deterministic symbolical combinatory (SC) model that has the form of graph as a branching tree , , .
However, realization of algorithm (10) in the form of software in inconvenient as it is necessary to generate the full set of permutations of elements of initial matrix Y. But this algorithm has the advantage that in it singular situations do not arise. Therefore for its practical realization in works ,  a special symbolical combinatory (SC) model for definition of minors (11) of initial matrix has been developed. Model has the form of graph as a branching tree  and represents a composition consisting of independent elements. It allows to realize in software parallel algorithms for parallel high-speed processing of flight information. In this case there is a possibility of its processing directly in onboard computers of aerospace objects in real-time mode of flight:
Such model has been applied for inversion of matrices (3), as well as matrices (7) used in operations of decoding using the Fourier method which generally also have bad conditionality. Programming of model (13) is simple because sections of graph are formed from components of ordered numerical sequence consisting of positive integers.
For increasing its accuracy in operations of multiplication of numbers in branches of the graph, a SC-model has also been used which is free from rounding mistakes. An example confirming its efficiency is below given. The product of five long numbers represented by columns of matrix (14) was being calculated. The intermediate result was found as a vector the components of which represent numbers with accompanying powers. The end result is the sum of elements of the vector. Such model has been applied in problem of polynomial approximation of experimental information based on operation of inversion of Hilbert matrix. It is considered the standard of bad conditionality and, consequently, the problem above the 10th order cannot be solved even with application of supercomputers. Application of SC-model has allowed to calculate the inverse matrix with 100% accuracy for 20th order matrix; a result which is given in .
Scientific Proceedings of Riga Technical University (1-9)
1. G. Burov. 2011. Symbolical Combinatory Models of Algorithms for Processing Flight Information at the Flight Test Stage of Aerospace Objects. Technologies of computer control. issue 43.
2. G. Burov. 2011. Optimization of Algorithms for Processing Flight Information in Flight Test Stage of Aerospace Objects. Boundary Field Problems and Computer Simulation. issue 46.
3. G. Burov. 2009. Principles of Automation of Processes of Identification of Aerospace Objects Characteristics at a Stage Flight Tests. issue 41.
4. G. Burov. 2007. Method of Functional Transformations in Algorithms of Computer Control of Analog Technical Objects. Technologies of computer control. 32 issue.
5. G. Burov. 2007. Models for Decoding the Results of Computer Control of Analog Technical Objects. 49 issue.
6. G. Burov. 2006. Modeling for Decoding the Results of Computer Control of Analog Technical Objects. 49 issue.
7. G. Burov. 2005. Formation of computing algorithms on the basis of graph address structures. Applied Computer Systems.
8. G. Burov. 2005. Combinatory models of inversion of special type matrixes, vol. 47.
9. G. Burov. 2004. Address computing models for tasks of identification. issue 46.
10. G. Burov. 2011. Symbolical combinatory models in problems of identification of analog technical objects. XI International scientific practical conference on high technologies. 2011. St. Petersburg, Russia.
11. L. Ljung. 1987. System Identification - Theory for the User. Prentice-Hall, Englewood Cliffs, NJ.