Functional control of the technical condition method for aircraft control system sensors under complete parametric uncertainty

The control system sensors failures can cause the aircraft stability and controllability deterioration. Such failures fast and reliable inflight detection and localization allows minimization their consequences and prevention of an accident. Direct application of traditional parametric methods for sensors health monitoring with the use of their mathematical models is impossible due to the lack of information about the real inputs on their sensitive elements. This leads to the need for the problem of aircraft flight dynamics modeling with a high level of uncertainties to be solved, which complicates the application of functional test methods and determines the necessity of excessive sensors hardware redundancy. Widely known nonparametric methods either require a prior knowledge base, preliminary training, or long-term tuning on a large real flight data volume, or have low selective sensitivity for the failed sensors reliable localization. This paper expands the application of the well-known nonparametric failure detection criterion, based on the analysis of the linear dependence of the input-output data Hankel matrix columns and solution of the sensor failures localizing problem. Necessary and sufficient solvability conditions are given, the structure and the criterion values are determined in an analytical form before and after the failures occurrence. The proposed method does not require functional or hardware redundancy, prior information about the parameters of mathematical models and their stability, identification, observation, or prediction problems solution. The efficiency of the method is shown on the Boeing 747–100/200 longitudinal model example. Fast tuning, fast response and selective sensitivity of the developed algorithms are noted.


INTRODUCTION
The necessity of aircraft operations safety improvement determines the relevance of developing algorithms which are able to detect onboard equipment and systems failures. Sensors direct and feedback aircraft control system connections failure, as a rule, cause changes in the structure of the aerodynamic relations of the aircraft, which may lead to deterioration of the aircraft stability and controllability characteristics. Rapid and reliable detection and localization of sensor failures in their technical condition monitoring process allows you to minimize undesirable consequences and take in-time measures to prevent an accident.
Parametric (model) methods (such as: filtration, observation, forecasting, identification, parity relations, redundant variables, graph-theoretic methods, etc.) [5][6][7][9][10][11] are the most widely spread methods and are considered to be the classical ones. These models either directly or indirectly utilize the real objects mathematical models parameters which values are set a priori based on the familiar physical operational guidelines or are evaluated during the identification process.
Direct use of parametric methods for monitoring the sensors technical conditionbased on their models is impossible due to the lack of information about the real signals input which their sensitive elements receive. This leads to the necessity of solution the aircraft flight dynamics modeling problem with a high level of uncertainty caused by non-linearity, unsteadiness, inaccuracy and nonidentifiability of mathematical models [9,12,18]. The resulting model errors inevitably cause an increase in the threshold values of the applied criteria, which increases the failures detecting and localizing time, it also reduces the reliability degree of the tasks to be solved. These problems impede the use of functional control methods and necessitate the use of flight parameters sensors hardware redundancy, which excess multiplicity is determined by the majority logic algorithms of the aircraft built-in control system.
Nonparametric methods do not require information about the parameters of the controlled objects models and are based on their input and output signals analysis measurements. Such methods are related to intelligent ones, since they consider the controlled object as a "black box" and make it possible to solve the problems for non-stationary and nonlinear systems under conditions of complete parametric uncertainty.
Widely known nonparametric methods based on knowledge (expert, neural network, genetic, fuzzy methods, support vector methods, etc.) [1,5,8,13,14], do not use explicit system of models, but require a prior knowledge base, prior training, or long-term configuration on a large volume of real flight data. Nonparametric methods, which are completely based on signal analysis (methods for analyzing Hankel matrices, principal and independent components, statistical, factor, and cluster analysis, partial least squares, subspaces of states, blind identification, etc.) [1][2][3][4][5][6][7][15][16][17][18][19], do not require any prior information about the object of control, while their configuration in real time requires a data preprocessing stage. These methods are characterized by high speed and reliability of failure detection, but they need additional transformations in order to localize them, since they have low selective sensitivity [15,17].
This work continues research in the field of aircraft avionics technical condition monitoring and diagnosing by means of nonparametric methods and expands the application of the well-known failure detection criterion based on the input-output data Hankel matrix columns dependence analysis [15][16][17], and solution of the control system sensor failures localizing problem. The scope of the work is limited by the deterministic discrete stationary linear mathematical models of the controlled objects with completely measurable conditions.

PROBLEM STATEMENT
Let the dynamics of the aircraft with the functioning flight control system be described by a linear discrete model in the state space by the vector-matrix "input-state-output" form where, x, y, u are vectors of states, measurements, and controls of dimensions n x , n y and n y respectively; A, B, C, and D are matrices of proper dynamics, control efficiency, measurements, and direct communication, respectively; and i-is the discrete moments of time.
Let's assume that at the moment of time f i i  there was a multiple simultaneous aircraft control system sensors failure type: where F -is the failure matrix are the elements that characterize the sensor calibration violations atf k ≠ 1≠ 0/ Only relying on the results of the control vectors u measurements and measurements y without having information about the model A, B, C, D parameters, it is necessary to determine the fact and time i f when the failure occurred, as well as to localize the measurement channels k in which the sensors failed.

FAILURES DETECTION PROBLEM SOLUTION
Let's consider the time interval before failures occur when The entire set of solutions with minimal parameterization is described by the expression where L Q , R Q -are left and right canonizers that formalize direct and inverse Gauss transformations, -is an arbitrary matrix.
Then, according to [6] while performing the condition of the equation solvability let us define the state vector for the current and previous time points taking into account (7)   which, according to (1) are also connected by the expression 1 1 .
Let's substitute (10) and (11) into (12) and combine (13) and (9) into one system of matrix equations and then multiply it on the left by the inverse matrix of the left canonizer Having completely measurable 0 R C  states, we can assume R C I  without loss of generality, and the inverse matrix in accordance with (8) takes the form of an identity the substitution of which into (14) leads to the expression which, with the account of the input-output matrix notation introduction will take the compact form of an equivalent (1)-(2) aircraft flight dynamics model of the "inputoutput" type

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After the failure occurs, expression (3) can be written in the similar way as expression (2) i where, ˆˆf Let us further assume that at the moment of time i we also know h-1 of the previous signals values. Then we can write models (15) and (16)

in matrix form
Any linear right-sided matrix equation of the WQ H  form with known matrices Q, H, is solvable with respect to an unknown matrix W if and only if the solvability condition is met [20] where = 0. The fulfillment of condition (20) guarantees the existence of linear matrix models of the (17) and (18) where h i Y  -is the errors measurement matrix caused by failures, so the performance of (20) is violated.
Therefore, the condition (22) can be used as a simple criterion for the aircraft control system sensors failures detection [15][16][17], the quantitative value of which for each moment of time is determined, for example, using the Frobenius matrix norm Criterion (23) will be zero before and after the failure occurs, when the control data matrices do not contain distorted measurements, and it will exceed an acceptable value when the control window includes the moment when the failure occurred.

THE SOLUTIONOF FAULT LOCALIZATION PROBLEM
We'll show further that the condition for detecting sensor failures (22), if considered line by line, can also be used to solve the problem of the failed measurement channel locating. To do this, we define the structure and values of the matrix measurement error in (21).
We need to notice that at the exact moment of failure at  (15) and (16): -is the number of measurements before and after the failure occurrence moment, accordingly; Then, taking into account (25), we can explicitly write the value of criterion (22) 1ˆ , in that case, the quantitative criterion for detecting and localizing the k sensor failure by analogy with (23) will have the form of where It should be noted that onwards during the for functional sensors in the general case may also differ from zero due to the availability (27), which's influence, as a rule, is tried to be eliminated while solving the problem of fault localization without taking into account (29) [15,17]. In particular cases when performing the identity when, for example, its own dynamics matrix is equal to zero, has a diagonal form, or the failure leads to scaling of all measuring channels signals by the same gain coefficient, condition (22) retains its necessity and sufficiency for the entire control time interval. Moreover, while solving the practical problems, as it will be shown later, the value of the (29) criterion for functional channels measurement, as a rule, is several orders of magnitude lower than for channels with failed sensors. This fact is determined by the characteristic structure of the right zero divisor of the input-output Hankel matrix data, which analysis requires further theoretical and practical research.

EXAMPLE OF THE PROBLEM SOLUTION
In

CONCLUSION
The result of the conducted research, shows that the failure detection criterion based on the analysis columns linear dependence of the input and output Hankel matrix system data can be used for reliable localization of the failed measurement channel right at the moment when the sensors fail. The structure and importance of the before and afterfailure occurrence detection criteria are defined analytically. The required and adequate conditions for the fault localization problem solution existence are given.
The proposed method of functional detection and localization of the aircraft control system sensors failures in flight is only based on the analysis of its regular input and output signals and does not require functional or hardware redundancy, solution, observation or forecasting of identification problems. It is not affected by model errors, since it does not require information about the parameters of the aircraft model, while, unlike similar nonparametric methods, it does not use logical or statistical calculations, training, or long-term configuration, and can be used to solve problems of technical condition control in the state of complete parametric uncertainty, even in cases of instability and non-identificability of the aircraft mathematical model. The efficiency of the method is shown by means of the heavy mainline aircraft longitudinal movement linear model example. Fast configuration of the developed algorithms, instant failures detection and localizationare noted, as well as high relative sensitivity of the criterion.
The Hankel matrices analysis methods can be used as a base for the unified mathematical apparatus of the hybrid active control systems synthesis [1,2,8], based on the joint use of parametric (analyzing the dependence of data matrix rows) and nonparametric (analyzing the dependence of data matrix columns) methods. The practical implementation of such a system will increase the level of fault tolerance of aircraft control system with a reduced multiplicity of its elements hardware redundancy. Исследование выполнено при финансовой поддержке РФФИ в рамках научных проектов № 20-08-01215, №18-08-00453, №19-29-06091