Elaboration of Evaluation Criteria for a Mathematical Model of a Fuel System in a Light Helicopter

Abstract

In this article, a calculation model for a fuel system of a light helicopter was developed for experimental and operational testing. The model analyzes the operation of not only the units, but also the whole fuel system, offering the basis for functional tests and failure detection. For an actual model-based design, it is necessary to ensure that the model conforms to a real object or process. This article addressed the problems of ensuring the accuracy of the calculation and the computational speed when developing a reliable mathematical model of an actual technological system using structural modeling methods. A system with evaluation criteria allows complex field tests to be replaced with validated calculation models.

1. Introduction

Current aviation equipment, particularly transport and passenger helicopters, must meet the highest requirements in terms of its safe operation [1,2,3].

The analysis of major incidents involving helicopters showed that although fire occurred in only 8.7% of cases, this caused 60.4% of all deaths. In addition, 78.5% of post-accident fires occurred due to damage to fuel cells and/or fuel pipelines [4].

In general, such accidents stem from hard landings, which are relatively common for helicopters. The use of new materials and design solutions for the fuel system offers a means of protecting machines from fuel spillage, ignition, or explosion [5,6].

Numerous systems solving various tasks are installed on board the aircraft. The fuel system of a light helicopter is designed for carrying the required amount of fuel on board the helicopter and for supplying fuel to the engines continuously under all operating modes.

Unlike the earlier fuel systems that included tanks and simple pipelines, the fuel systems installed on modern helicopters comprise complex interconnected subsystems [7]; therefore, it is necessary to prevent possible emergencies and problems during operation.

The fuel system of a helicopter includes the following main subsystems:

• The fuel stowage subsystem;
• The fuel distribution subsystem;
• The fuel measurement and control system (FMCS-AS).

The fuel stowage subsystem ensures the stowage of the required amount of fuel on board the helicopter, along with the interconnection of the ullage space of the fuel tanks with the atmosphere through the drainage lines. The stowage subsystem includes:

• Fuel bladders (front left feeder, front right feeder, and rear tanks) with embedded parts (mounting slabs, access holes, deformable inter-tank connections, and pipeline outlets);
• Motor-operated dump valve (one in each fuel tank) and two accident-tolerant dump valves in the rear fuel tank;
• Drain system, including flexible drain tubes, bypass accident-resistant drainage valves, and flame arresters.

The fuel distribution subsystem pumps fuel to the engines and from the rear to the feeder tanks, along with refueling the tanks. The pumping system is designed to pump fuel to any engine from any pump, block supply lines in the case of fire or engine failure, and supply active fuel to the transfer pumps.

The fuel distribution subsystem includes:

• Booster pumps equipped with a pressure sensor with a bypass following pump;
• Flexible tubes of pumping line;
• Check valves;
• Double-drive fuel emergency shutoff valves;
• Cross-feed valves;
• Thermal-relief valves;
• Fire-resistant hydraulic hoses.

While improving aircraft complexity, the issue of establishing a mathematical model to facilitate the calculations and reduce the time and cost of the work is addressed more and more often. The methods of mathematical modeling are widely used, with a tendency to reduce the number of field tests during design, which allows the behavior of the simulated component in a helicopter system to be monitored when it is impossible to carry out a full-scale experiment either from a physical standpoint or due to limited time and capital resources.

Generally, when developing a model of a technological system, it is necessary to ensure that this system meets all the requirements. However, how does one verify that the implemented system corresponds to the designed model that satisfies the requirements of the verified specification?

This article discusses the criteria for verifying whether the requirements of a technical specification are met when developing dynamic models of technological systems. Considered are the requirements that can be expressed numerically and verified mathematically by means of a specific computational model. Although only several general requirements can be evaluated, their verification is highly time- and money-consuming when developing dynamic models of an object. Such criteria should ensure that the mathematical model is tested as per the aviation rules.

When describing technical requirements, several criteria can be distinguished, each calling for different approaches for automatic verification.

Therefore, in this work, the developed custom model of the aircraft fuel system is tested, along with the determination of the criteria for validating the model. The main input parameters and their validation have the most pronounced effect on the accuracy of the developed model. Although many parameters can be taken from the specification, it is recommended to obtain them by measuring and testing.

2. Testing of a Light Helicopter

The main stages in the life cycle of a complex technological system include research and development, full-scale production, and operation, and the testing comprises an essential science-intensive stage.

Based on aviation regulations, the certification basis of aviation equipment requires that fuel is supplied at a flow and pressure that ensure the normal operation of the engine units under all operating conditions and that prevent air from entering the system [8]; this must be demonstrated through tests recognized by the commission. The tests should be carried out using a test bench that reproduces the performance characteristics of a fuel system. The fuel supply to the engines should ensure 100% of the airflow demand for each operation mode of an aircraft at the required input pressure and temperature of the fuel.

It is necessary to verify the minimum fuel and the functioning of the primary and standby pumps. The performance of a helicopter system in a hot environment at a fuel temperature of at least 45 °C in the engine supply lines is examined, along with the absence of steam–air locks in the pipelines and the excessive vaporization of fuel, in order to confirm the compliance to the requirements of the aviation regulations.

As the reliable operation of the engines ensures the safe flight of any aircraft, it is necessary to provide an uninterrupted supply of required fuel at the specific pressure and temperature during operation in various modes, including various movements in space, accompanied by changes in the bank and attitude angles.

It is important to study and analyze various failure situations during the operability testing of a fuel system during long-term operation. The mathematical model can facilitate the analysis and verification of the many operating modes of the fuel system of a light helicopter.

3. Review of Existing Mathematical Models

In the article in [9], the authors addressed the development of helicopter flight dynamics, based on the simulation of an aircraft with a fixed wing, by studying a nonlinear system which was replaced by the analysis of a linear system or its equivalent aircraft system.

This method offered the necessary control characteristics for a helicopter that can be used for the initial design of its control system. However, a linear model can describe the flight dynamics of helicopters characterized by little maneuvering ability.

In the articles in [10], the authors considered flight dynamics, along with the main rotor, rear end, and propulsion unit, as well as coasting flight.

Generally, mathematical models of a helicopter can be divided as follows:

• Those for theoretical analysis;
• Those for numerical simulation;
• Those for real-time simulation.

In the article in [11], a mathematical model comprised a design basis for evaluating the flight characteristics of a helicopter. Motion, inertial resistance, and aerodynamics were taken into account, as well as the nonstationary and nonlinear characteristics, in order to provide the physical principles and mathematical expression for each part of a helicopter. The possible improvement of the helicopter configuration was considered, with the aerodynamics of the rotor and the aerodynamic interaction between the rotor, fuselage, and other structural elements being modeled. The present simulation method involved the determination of the flight profile, followed by the loading of the control input data during maneuvering. Although useful in studying the maneuvering flight profile of a helicopter, in practice it proved ineffective as it did not take into account the operation of the engines.

A code describing the flight dynamics for the light helicopter Helium 2 [12], developed at the University of Maryland, included a trailing vortex model to improve the accuracy of the calculation. In addition, it was possible to carry out a calculation with several rotors. In this study, the necessary parameters for the various configurations of the aircraft were calculated. The variations in the back angle of a helicopter and an aircraft with an inclined propeller were also compared. The flight dynamics model can be adjusted using empirical formulas to improve accuracy during stable flight and during flight maneuvers at a low amplitude. Aerodynamic characteristics are crucial for maneuvering, especially for modern rotary-wing aircraft. The program code for the Helium 2 helicopter exhibited better control accuracy. Although this indicated increased accuracy in the calculations of the wake of the rotor and other parts, it was, however, lower than that of the model obtained in GEN HEL [13] due to the significant aerodynamic interference inside the axial rotor system. Hence, the method of calculating the rotor wake used in Helium 2 (the free wake model) may lead to additional errors.

The article in [14] described a simulation of the aircraft fuel management system, studying the logic and sequence of its operation for both normal flights and in cases of malfunctions. The airborne control and reconfiguration program of the aircraft fuel system equipped with six tanks was tested.

In the work in [15], a tool based on an adaptive neuro-fuzzy inference system (ANFIS) was proposed for the evaluation of the fuel system of a small aircraft. The model was designed to identify malfunctions present in the fuel system of the aircraft, as well as to prevent them. The ANFIS tool used sequences that are determined according to the fuel consumption by the engine and in the tanks. In addition, this software learns by using the previous fuel consumption value.

In the study in [16], methods of online monitoring and analysis of the aircraft fuel transfer system were used. The diagnosis and fault detection based on the (MBD) model was addressed to monitor the system behavior and to recognize inconsistencies between the predicted and observed values.

Software for the virtual simulation of the aviation prognosis and health management (PHM), developed by Shen Ting, Fangyi Wan, et al. [17], represents an integrated approach based on the modeling method and the signal processing method. The fuel system was controlled by the center of gravity (CoG), which is essential for flight safety. Virtual simulation based on “fault trees” was deduced on the basis of possible fault states determined during an undesirable state of the system. The limitation of such a model is associated with an incorrect initial assessment or process model.

In the work in [18], modeling at low pressure and a comparison with experimental studies were addressed. A model of the polarization curve was determined that can describe its changes when operating conditions change. The predictive behavior of the model was investigated using specific polarization curves.

Several authors attempted to use the results of experiments to simulate fuel cells. In these works [19,20], the influence of factors on the response time of the unit relative to a given power was estimated in order to assess those influencing the system performance. In addition, possible optimizations in the operating conditions were determined to increase power.

In the work in [21], a two-level factorial design for three factors was investigated: temperature and hydrogen and air humidity at the inlet. A semi-empirical model composed of six parameters was created. The model used on an experimental database in order to assess the impact of an individual factor on the performance; however, this was valid only for a limited circle of specific models.

In the article in [22], the authors described a system of integration and the testing of units within an aircraft fuel system, which allows for the experimental detection of malfunctions. The simulation model of the system created using the AMESim software v2019.1 was validated for refueling under pressure on the ground. The operation of the refueling control valve determines the correctness of the system operation.

In the paper in [23], the authors presented a numerical study of a bird colliding with a helicopter tank using fluid analysis. The fuel spillage was estimated using the Euler–Lagrange method and hydrodynamic calculation.

In the work in [24], the command profile of the helicopter controller and the trajectory tracking controller based on a simplified model were proposed. Two contours were considered, where the inner contour allows the orientation of the helicopter to be followed and the outer contour uses performance monitoring to ensure that the error at the outlet of the position controller converges. The position/orientation tracking method based on neural networks was studied in the work in [25].

In the article in [26], the flight control systems of an unmanned aerial vehicle were modeled. The developed model was evaluated and refined on the basis of the parameters of the throttle shutter, the thrust coefficients, the torque, and the moment of inertia. Experimental coefficients and parameters were determined in the MATLAB/Simulink model using the Newton–Euler equations. The flight control system in the horizontal plane and vertical direction was modeled by analyzing the reaction of the system. The model showed good convergence of the response of the mathematical model with the response of the actual model.

In the work in [27], a formulated conjugate problem statement for aero gas dynamics, internal heat and mass transfer, and the thermal strength of aircraft structures was solved using numerical modeling based on the iterative solution of three problems.

The work in [28] addressed a distributed control of fluids in aircraft, focusing on the fuel system, which allows the amount of aviation fuel to be measured when refueling aircraft. The model accounted for three modes of operation: refueling under pressure and on-engine fuel supply.

The model of a helicopter fuel system developed by the authors of this article differs from those discussed above in that the operation of the whole system can be analyzed, giving a basis for functional tests as well as the detection of various malfunctions. Estimates for applying an integrated approach are also included in the system.

4. Structure of a Mathematical Model of a Helicopter

The developed simulation model of a helicopter fuel system includes the complex interaction of fuel and system units, the environment, the helicopter structural elements, the engine operating modes, and the power supply system. Here, the entire complex model should be dynamic. The more precisely the operation of the various modes and units in the whole system is implemented in terms of flight elevation, the more reliable the result of the study and the output parameters. A schematic diagram of the custom mathematical model of the helicopter fuel system is shown in Figure 1.

The model of a light helicopter includes two main parts: an automation diagram and a thermal hydraulic diagram, as is shown in Figure 2.

The implemented computational thermohydraulic code is designed to calculate the dynamic behavior of the main parameters characterizing a compressible and incompressible heat transfer agent in the thermohydraulic loops of arbitrary topology. Thermohydraulic schemes solve the equations of mass, momentum, and energy conservation for a fluid (in a one-dimensional, one-velocity approximation), as well as the non-steady heat-transfer equations for thermal structures (channel walls), based on a one-dimensional, non-steady homogeneous model of the incompressible or compressible flow [29].

The thermohydraulic scheme offers a mathematical apparatus for modeling the nonstationary flow of a heat transfer agent in loops of arbitrary topology, including heat exchange with the tube walls and ambient environment and the operation of the valves and turbopump equipment.

The combination of nodes and channels connected by flow lines comprises the main part of the design scheme, which is identical to the basic technological scheme of the simulated system.

The thermohydraulic loop and the automatic scheme are connected through a database of signals, which communicate with a file containing the structured global signals of the project.

The signal database allows for a joint calculation of several projects with a data exchange via RAM on one computational node or a network on several computational nodes. In addition, the database supports the mechanism of the initial states, filtering signals and organizing queries to the database from the programming language.

Thus, the dynamics of the model are primarily determined by the automatic scheme, i.e., by the variation of the signals that are formed in the diagram window of the automatic scheme and transmitted to the thermohydraulic loop. Such signals include variations in flight altitude, helicopter speed, opening/closing valves, turning on/off pumps, etc. An example of the scheme is shown in Figure 3.

The developed mathematical model of an accident-tolerant fuel system (ATFS) provides the following functions:

• Airborne fuel stowage;
• Uninterrupted fuel supply to the main engines at a pressure sufficient to ensure stable operation of the engines under all expected operating conditions;
• Fuel supply to the engines, including during failure operating modes (failure of one of the engines, failure of one or two backing pumps);
• Single-point refueling under pressure;
• Refueling through filler necks by gravity;
• Controlled fuel draining from tanks by gravity;
• Fuel tank drainage during refueling and fuel use;
• Management and control of the operation of an accident-tolerant system.

5. Methods for Elaborating Evaluation Criteria

In order to be successfully applied to the test of a technical system, along with the full-scale tests, it is necessary to verify a mathematical model according to specific criteria. General approaches to establishing such criteria and the methods of their possible mathematical and software implementation are described in the works in [30,31].

The criteria for the adequacy of a mathematical model can be conditionally divided into two groups:

• Goodness of fit;
• Reliability of modeling results.

The first group includes criteria that are used to evaluate the performance of a new model based on a comparison with the previously verified mathematical models, as well as the full-scale tests of the various aggregates of the model. For example, if it is known how an individual unit or the whole system should respond to variations in test conditions (temperature, pressure, or operating mode), a criterion for the compliance of the behavior associated with the tested model with the expected results is established. Identical conformance inspections can also be carried out in order to simulate possible failures if the test data of the simulated system are available for these modes.

In the case of the significant deviations of the model parameters from the expected values, it is necessary to address the sources for such deviations and, if necessary, correct the mathematical model.

For these criteria, the numerical functions are optional since it is crucial to establish the fundamental behavior of the model under all operating modes. Therefore, such criteria can be considered as logical variables (for example, if it is known that the pressure in the tank should increase with an increase in the ambient temperature, the corresponding variable should take the value “yes” if this occurs, and “no” otherwise).

It is necessary to develop special functions for the criteria of the second group which take into account the relationship between the combination of the defining requirements (external parameters, such as verification), along with the characteristics of the system that are determined or refined during the modeling process (internal parameters). By using these functions, a system of limitations for the external and internal parameters in the form of equations or inequations can be established that allows the range of some parameters to be determined at the fixed values of other parameters for various calculation modes [31].

The operating modes that can be evaluated using the developed mathematical model include:

• The refueling of fuel tanks;
• The nominal operating mode;
• The failure of an electric submersible pump;
• The failure of two electric submersible pumps;
• The flight following profile;
• Engine failure;
• Fuel draining;
• The assessment of the all operating modes of a fuel system in the case of failures in the fuel drainage system.

As an example, Figure 4 shows the results of a dynamic calculation under the refueling mode. When the target level is reached, the flow passage is blocked by a float valve so as to finish the tank refueling; furthermore, upon pouring the fuel into neighboring tanks, the fuel level decreases, followed by the subsequent opening of the flow passage. The ventilation during refueling under pressure is ensured by a fuel drainage system. The algorithm simulating the float valve controls the fuel level in the tank (for continuous filling, it is recommended to refine the float valve model).

The algorithms for controlling the units and engines of a fuel system are based on the processing of the incoming data arrays, given that the operation of the pumps and engines is based on the consumption characteristics. The simulated failure system describes the behavior of the fuel system under various conditions. A mathematical model, along with the control parameters, can be adopted to refine and design various systems.

Figure 5 shows an example of the control panel of the model in the initial state for calculating the FS under the failure mode of one electric submersible pump (in the right tank). In the case of an emergency, if one of the fuel pumps of the engines fails, the cross-feed valve opens, with the fuel flowing to the engine via the cross-feed line. The simulation result is shown in

Table 1. Simulation data.

Parameter	Value
Fuel consumption by electric submersible pump R (in right forward tank), kg/s	0.15
Fuel consumption per engine (right), kg/s	0.0556

.

Therefore, the validation criteria for the mathematical model of the helicopter fuel system should include numerical estimates for comparing the parameters of the new system with those of the test (numerical or full-scale) results. From a mathematical point of view, such criteria are formed by calculating the residuals of the various parameters of the tested and “reference” model at all possible points. It is recommended to express these residuals as dimensionless values, such as the percentage of deviation from the “reference” values, in order to establish an objective function for a comprehensive assessment of the adequacy of the model [30]. The following objective function was adopted for the developed model:

$$F={\displaystyle \sum }_{i=1}^n{\lambda }_i\times \left|\frac{p_i-P_i}{p_i}\right|; {\displaystyle \sum }_{i=1}^n{\lambda }_i=1$$

(1)

where ${\lambda }_i$ is the expert evaluation of the significance (weighted coefficient) of the i-th parameter of the model; $p_i$ is the value of the i-th parameter of the tested model; and $P_i$ is the value of the i-th parameter of the “reference” model.

The objective function should be minimized at the first stage of the verification of the mathematical model to be further used for a comprehensive assessment of the model operation under various calculation modes. In order to obtain reliable results, the points where the sensors for the various parameters are installed in an actual system (pressure sensors in tanks, temperature and pressure sensors in tubes, heat transfer agent flowmeter, etc.) should be taken as testing points.

The final verification of the mathematical model is carried out following the bench and flight testing of the system. Based on the results of such assessments, a database can be created for clarifying the weighted coefficients of the objective functions when developing new models of similar technological systems [30].

6. Evaluation of Mathematical Model

As an example, the helicopter refueling system is considered; it includes units for refueling under pressure: a pressure refuel connection, a solenoid valve, and a float valve. It is possible to carry out refueling by gravity or under pressure. The initial data for calculating the mode specified in

Table 2. Initial data.

Parameter	Value
Refueling time, s	600
Centralized fueling pressure, kgf/cm2	5
Refueling capacity of rear tank, L	405.3
Refueling capacity of right forward tank, L	226.77
Refueling capacity of left forward tank, L	226.77

are loaded into the developed mathematical model.

The simulation results are shown in

Table 3. Simulation data of the refueling mode.

Parameter	Value
Refueling time, s	411
Centralized fueling pressure at tank inlet, Pa	489,220
Average speed, m/s	3.5
Refueling capacity of rear tank, L	404.1
Refueling capacity of right forward tank, L	225.3
Refueling capacity of left forward tank, L	226.3

.

Figure 6 shows the graphs of the time-dependent fluid pressure in the tanks when modeling the refueling process at the blocked drain line. Given a safety valve, the pressure is distributed evenly, with the pressure inside and outside the tanks being equalized. The figure shows the time-dependent volume of the gas in the fuel tank; the gas layer decreases with an increase in the fuel level.

As a second example, let us consider the operation of the system under a normal fuel supply by the fuel pump in order to evaluate the operation of the custom mathematical model. Pumping from the rear to the front feeder tank is achieved by two jet pumps installed in the rear tank. The active fuel for the jet pumps is supplied from the fuel pumps of the fuel supply system to the engines. The jet pumps ensure total fuel use in the rear tank.

It is worth noting that the flight profile is conditional; the analysis of the flight profile shows that it is possible to model the dynamic modes following the profile, taking into account the variations in the engine parameters and temperature corrections depending on the ISA (the possibility of modeling “hot” and “cold” days), as well as the failure situations during flights. The results of the profile flight mode are shown in Figure 7. The following characteristic parameters of a flight are taken into account: air flow rate, braking temperature, recovery temperature, atmospheric temperature, moisture content, and braking pressure [26], which makes this model universal for use beyond a helicopter flight profile.

Here, the fuel system should supply fuel at a mass flow rate of at least 200 kg/h and an excess pressure of at least 30 kPa at each engine inlet under all operation modes, as per the aviation regulations. The maximum excess fuel pressure at the start-up and operating modes should be less or equal to 150 kPa.

The graphs of the absolute pressure in the left and right engines under the refueling mode are shown in Figure 8. Given that the pump pressure is taken as 101,325 Pa, the excess pressure during refueling in the left and right engine lies within the range from 30 kPa to 150 kPa. The pressure values vary uniformly, with minimal deviations relative to each other.

Figure 9 shows the graphs of the absolute pressure dependences in the engines; the flight profile is modeled with an altitude excursion from 0 to 6000 m, as follows: ground parking from 0 to 30 s, takeoff upward to 6000 m from 30 to 770 s, flight at an altitude of 6000 m from 770 to 1500 s, and landing followed by parking from 1500 to 2600 s.

At the first stage of validation, the control values of the selected parameters were formed on the basis of an alternative mathematical model previously used to simulate the operation of the fuel systems of various aircraft [30]. In order to validate the developed mathematical model of the fuel system of a light helicopter, the following controlled parameters and corresponding weighted coefficients of the objective function were selected and are shown in

Table 4. Weighted coefficients of objective function.

Parameter Name	Designation and Units of Parameter	Weighted Coefficient, λ
1. Mass of fluid in rear tank	P1, kg	0.1
2. Mass of fluid in left tank	P2, kg	0.1
3. Mass of fluid in right tank	P3, kg	0.1
4. Fluid pressure in rear tank	P4, Pa	0.1
5. Fluid pressure in left tank	P5, Pa	0.1
6. Fluid pressure in right tank	P6, Pa	0.1
7. Fluid pressure at an inlet to left engine	P7, Pa	0.2
8. Fluid pressure at an inlet to right engine	P8, Pa	0.2

:

An example of a comparative graph of the pressure changes at the inlet to the left engine for two different models is shown in Figure 10.

As a comparison, identical operating modes of the system were selected:

• Refueling mode;
• Flight mode following selected profile on a hot day;
• Flight mode following selected profile on a cold day.

As both models can be run in real-time, the following algorithm was used to calculate the final values of the validation objective function F:

• Prior to the computational cycle, the value F = 0 was set;
• At each time step, a new value of the objective function $F\left(t\right)={{\displaystyle \sum }}_{i=1}^8{\lambda }_i·\left|\frac{p_i-P_i}{P_i}\right|$ was calculated and was compared with the current F value;
• When the condition $F\left(t\right)>F$ was met, the current value of the objective function was changed to a new value, with the corresponding time point being recorded.

Therefore, following the termination of the calculation, it was possible to control not only the relative deviations of the parameters in the compared models, but also the time when these deviations were the highest.

The validation results are shown in

Table 5. Simulation data.

Simulated Operation Mode	Value of Objective Function, F	Time of Highest Deviation, s
1. Refueling of fuel system	0.042	243
2. Flight following selected profile on a hot day	0.037	2053
3. Flight following selected profile on a cold day	0.034	2061

.

7. Conclusions

It is necessary to improve the mathematical methods for evaluating the algorithms in the research and development of the simulation mathematical tools.

A number of nontrivial problems should be solved while developing such a mathematical model. By designing the described model, a software framework can be established, which, if further used and improved, can significantly reduce the time and cost of manufacturing new products and, hence, increase the profits of the enterprise.

The mathematical model of a fuel system of a light helicopter described in the article allows various operation modes to be calculated (basic modes: normal operation, failure situations, refueling, a connection of drop tanks, and a profile of the flight).

The performance of the developed model was tested by comparing it with an identical model using a specific objective function. Here, the deviation of the resulting key parameters (within the range of 0.034 to 0.042, i.e., 5%) provides the necessary accuracy for further calculations and the analysis of the obtained data.

Simulated in the mathematical model, the failure system describes the behavior of the fuel system, including the operation of the units under various conditions. The algorithms for controlling the units within the fuel systems and engines are based on the processing of multiple incoming data.

As the described mathematical model of the fuel system displays compliance when simulating various operation modes, as per the aviation regulations, together with the control parameters, it can be used for further optimization by comparing the output data with the test results of the various units and flight systems, as well as for various functionality studies. A given objective validation function can be solved by the described mathematical optimization.