SIMPLIFIED MATHEMATICAL MODEL FOR AIRCRAFTS RESPONSE CHARACTERISTICS Constantin ROTARU , Oliver CIUICĂ, Eduard MIHAI, Ionică CÎRCIU, Radu DINCĂ… [626595]

SIMPLIFIED MATHEMATICAL MODEL FOR
AIRCRAFTS RESPONSE CHARACTERISTICS

Constantin ROTARU , Oliver CIUICĂ, Eduard MIHAI, Ionică CÎRCIU,
Radu DINCĂ

*”Henri Coandă” Air Force Academy, Brașov, Romania
(rotaru.co [anonimizat] , oliverciuica @yahoo.com , [anonimizat] ,
[anonimizat] , [anonimizat] )

DOI: 10.19062/2247 -3173.2016.18.1.7

Abstrac t: Human performance modeling provides a complementary technique to develop
systems and procedures that are tailored to the pilot’s tasks, capabilities, limitations and also,
offers a powerful technique to examine human interactions across a range of possible operating
conditions. From an initial review of past efforts in cognitive modeling, it was recognized that no
single modeling architecture or framework had the scope to address the full range of interacting
and competing factors driving human actions in dynamic and complex environments.
The paper presents two mathematical compensatory models , based on the aircraft dynamics
characteristics and frequency responses .

Keywords: aircraft dynamics , Laplace transform, frequency response.

1. INTRODUCTION

Hum an performance models were developed and applied to flight operations in order
to predict errors and evaluate the impact of new information technologies and new
procedures on flight crew performance. The usefulness of the human performance
modeling to the d esign and evaluation of the aircraft technology is determined by the core
capabilities – visual attention allocation, workload, crew interactions, procedures,
situation awareness and error prediction [1]. The modeling efforts revealed that human
performan ce models, even those cognitive architectures that have traditionally been used
in the context of psychological laboratory experiments, can indeed be useful tools for
complex, context dependent domains such as aviation. Specifically, the tools can be used
to address the design and evaluations of aviation displays, procedures and operations.
These models can be used to inform display design and the allocation of information
so as to optimize efficient scan patterns and increase the uptake of relevant informa tion in
a timely manner. Although the analysis and understanding of the airplane as an isolated
unit is important, for many flight situations it is the response of the total system, made up
of the human pilot and the aircraft, that must be considered. Many tasks performed by the
pilots involve them in activities that resemble those of a servo control system, so, the pilot
can be modeled by a set of constant coefficients linear differential equations [2, 3 ]. Much
of research in the field of human pilot descr ibing functions hasconcentrated on the pilot’s
performance in a single degree of freedom compensatory tracking task with random
system inputs, where the pilot controls a single state variable through the actuation of a
single control. A compensatory displa y is one in which the tracking error is presented,
regardless of the source of error.
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2. PILOT MODEL

Due to the complex nature of the situation it is possible to model the pilot in many
ways and to measure the model by employing a variety of techniques. O ne of the most
successful approaches to the measurement problem utilizes power spectral density
measurements of signal circulating in the control loop. The human pilot could be replaced
by a mathematical model consisting of two parts (Fig. 1): the linear d escribing function
(written in Laplace transform notation), ( )s Y, and the remnant, ( )t n. Since a linear model
is never able to describe the pilot completely, ( )s Y is insufficient by itself, and it is
necessary to include the remnant ( )t n, which is the signal that must be added in order to
have all the time sign als circulating in the system [4 ]. The ( )s Y selected to describe the
pilot in any particular task is chosen so as to minimize that part of the input signal to the
aircraft which arises from ( )t n. Thus, the linear pilot model that results is that which
accounts for as much pilot input to the aircraft as possible, and a measure of its adequacy
is the fraction of the pilot input to the aircraft accounted for by ( )s Y.

(a) (b)
FIG. 1. Linear model of the pilot -aircraft system (a) and displayed variables (b)

According to the Fig.1, the pilot must control the aircraft response ( )t m in such a
fashion that is matches as closely as possible the desired aircraft response. The pilot does
this by viewing the instantaneous error ( )t e and altering his input to the aircraft (Fig. 2).
It is found that the pilot’s control technique is influenced by the type of input ( )t i, the
dynamics of the control system, the type of display and the dynamics of the aircraft.
Although ( )t e is available in both cases, only the pursuit display separates the error into
its components and conveys this information to the pilot. The single degree of freedom
tracking task with a pursuit display is identical to the compensatory task, except that the
displayed variables are different. In the co mpensatory task only ( )t e are displayed,
whereas in the pursuit task both ( )t i and ( )t m are separately displayed. The data that
pilots utilize to fly can be found on cockpit display panels and qualitati vely different from
the cues used in correspondence judgments (Fig. 2 ). They are data rather than cues in that
they are precise, reliable indicators of whatever they are designed to represent. In the
electronic environment of the cockpit, the primary task of the pilot is to supervise and
monitor systems and information displays to ensure consistency of the information and to
restore them when disruptions occur.
All the possible airspeed and altitude conditions of an aircraft are visualized in the
flight en velope, where every aircraft performs the specific mission for which it has been
designed. Within this flight envelope, desirable flying qualities are defined as a
combination of characteristics both in terms of piloting the aircraft as well as in terms of
the aircraft response in itself [5, 6 ]. Specifically, this includes the analysis of three
factors: control authority (the pilot’s capability to generate appropriate aerodynamic and
thrust forces and moments), pilot workload (the physical effort while the pilots applies ( )t o ( )t n
Pilot model ( )s Y
Aircraft
model ( )t m ( )t i
Moving
input
symbol
Moving aircraft
attitude symbol ( )t e ( )t i ( )t m
Reference
line
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AIR FORCE AND AEROSPACE ENGINEERING

physical forces through its arms and feet to the control commands in the cockpit of the
aircraft), and pilot compensation (the mental effort by the pilot while at the co ntrols of the
aircraft) .

(a) (b)
FIG. 2 .Aircraft cockpit (a) and human pilot workload (b)

The aircraft dynamical equations are nonlinear in the inertia terms and in the
kinematical variables. The external force, especially the aerodynamic one, may contain
inherent nonlinearities. In the automatic and powered control systems used in aerospace
vehicles, there occur nonlinear control elements such as limiters, switches, and others.
The human pilot is the ultimate in time -varying nonlinear systems.

3. AIRCRAFT CONTROL CHARACTERISTICS

Two control paths that the pilot emplo ys for compensatory and precognitive control
were considered. For compensatory control, the pilot observes errors between the desired
and the actual response of the aircraft and applies the control to reduce or eliminate the
error. If the pilot has the ability to observe the task command directly, the aircraft can be
controlled to immediately follow that command without waiting for the error to develop.
With a priori knowledge of the aircraft behavior, the pilot can shape these control
commands to achieve the desired response.
Two models of the pursuit task are presented in fig. 3 , where the pilot is represented
by a pair of describing functions ( ) ( ) ( ) s Y s Y2 1, or ( ) ( ) ( ) s Y s Y4 3, since the pilot is considered
to have two inputs and one output. The function ( )t grepresents the turbulence acting on
the controlled aircraft.

(a) (b)
FIG. 3 .Compensatory task with pilot model (a) and the model of the pursuit task (b)

The response of the aircraft to either control inputs or di sturbances may be represented
by the transfer function of the aircraft with respect to the controls or by the transfer
e
( )t n
Pilot
model
( )s Y2

Aircraft
model ( )t m ( )t i

( )s Y1
( )t g
( )s A
( )t o
( )t n Pilot model
( )s Y3
( )s A
( )t m
( )s Y2

( )s Y4 ( )t g
( )t e

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function of the aircraft with respect to the disturbances. The control input is defined both
by the pilot’s open -loop inputs in response to the command and the pilot closed -loop
inputs in reaction to the error between the command and the aircraft’s actual response.
The first element of the control system that the pilot encounters is the control inceptor,
namely, stick, pedals, throttle and thrust -vector levers. Nonlinear characteristics typical of
most mechanical control devices must be considered, including hysteresis, breakout force
and changes in the force gradient with control deflection.

4. HIMAN PILOT MATHEMATICAL MODEL

To carry out analyses of the aircraft’s flying qualities, it is necessary to have a
mathematical representation of the pilot. Pilot models are typically represented in transfer
function form that relate the pilot’s control output in response to perceived error in the
aircraft’s response compared to the desired command,
1 11
+ ⋅⋅+ ⋅+ ⋅=⋅ −
s Te
s Ts TK Y
Ps
IL
p pτ
(1)
The first element of the transfer function is the gain,pK, that determines the amount
of control the pilot commands in proportion of the perceived error. The pilot can also
perform dynamic compensation such lead (LT- lead time constant) and lag (IT- lag time
constant), as indicated in the equation (1). Elements that cannot be adjusted are associa ted
with the transport delay, τ, involving visual observation and mental processing of the
information. The human muscle structure cannot respond instantaneously to command to
move and exhibit a lag in response, which is represented b y the term pT in the equation
(1). The observed variation of the pTwith forcing function bandwidth ranges from
s1 . 0to s6 . 0. The time delay represented by the se⋅ −τ term is due sensor excitation,
nerve conduction, computational lags and other data processing activities in the central
nervous system. The parameter τ is considered constant because it appears to be invariant
with forcing function an d controlled element dynamics for either single or dual random –
appearing inputs tasks. The representative values for time delay are of the order of
s2 . 0=τ and the time lag pT is approximately s1 . 0.
The expression ( ) ( ) 1 / 1+ ⋅ + ⋅s T s T KI L p represents the major element in that adaptive
capability of the pilot which allows him to control the dynamic devices. Its function is the
modification of the stimulus signal into a suitable man -machine system operation.
The form of the pilot transfer function shown in the equation (1) was identified from
aerospace publications and laboratory measurements acquired from human subjects. Pilot
workload consists of both the mental and physical effort required to control the air craft to
achieve the desired response. Mental effort, which is difficult to quantify, is associated
with anticipation required to generate lead to compensate for poor aircraft response
characteristics. Physical workload can be described as the work the pil ot must expend in
moving the control inceptors against their resisting force. An example for an open -loop
bank angle response to aircraft lateral control is
( )( ) 11
1+ ⋅⋅+=s T s sLs A
AAδ (2)
where
ALδis the lateral cont rol sensitivity, RT is the roll mode time constant and AT is the
control surface actuator time constant.
For the longitudinal aircraft control, the simplified system of equations has the form
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AIR FORCE AND AEROSPACE ENGINEERING

( )( )
( )( )sMZ
ss
sMs M s MU s Z U s
e
qeeδθα
δδ
α αα


=
⋅



− + −⋅ − − ⋅
21 1
 (3)
where s is the Laplace transform parameter. The other variables in the above equation are
the longitudinal control derivatives, the pitch and attack angles ( θand α) and the
elevat or command eδ [7, 8].
In order to obtain the dynamic response characteristics of the human pilot, we have
made laboratory measurements acquired from human subjects, for control forces of the
stick, pedals, throttle and aircraft command surfaces. We have estimated the time delay,
the lead and lag time constants of the pilot model represented in transfer function form
and also we built up the mathematical model of the whole system “pilot and aircraft”.

5. NUMERICAL RESULTS

The impulse an d step responses are calculated for an aircraft with the following
parameters: wing span, m8; mean aerodynamic chord, m5 . 1; sweep angle, 18; mass of
aircraft kg 3000 ; xxI moment of inertia, 21000 mkg⋅; yyI moment of inertia,
26200 mkg⋅; zzI moment of inertia, 26800 mkg⋅; xzI product of inertia, 2200 mkg⋅;
flight Mach number, 0.7. Some numerical results are presented in the following figures.

(a) (b)
FIG. 4 .Impulse response (a) and step response (b) for the aircraft elevator command

(a) (b)
FIG. 5 .Polar plot of pilot -aircraft transfer function in late ral (a) and longitudinal (b) movements
SCIENTIFIC RESEARCH AND EDUCATION IN THE AIR FORCE-AFASES 2016
59

The dynamic characteristics of the integrated “pilot -aircraft” system were studied with
the frequency -response method. The middle line (red color) in f ig. 5 corresponds to the
following parameters: 1=pK ; 2=LT ; 5 . 0=PT ; 15 . 0=τ .

CONCLUSIONS

The assessment of handling or flying qualities of airplane depends on pilot skills.
When the pilot flies an aircraft he forms subjective opinions concerning the suit ability of
the man -machine system for performing the assigned task. In arriving at an assessment he
is influenced by many parameters. These range over a wide spectrum and include the
response to external disturbances, the ease with which instruments can be read, mission,
visibility, weather conditions and the familiarity of the pilot with the present aircraft and
mission. To be able to asses aircraft handling qualities and human performances, one
must have a measuring technique with which any given aircraft characteristics can be
rated. If the pilot is expected to assume manual control the system should be structured
that he is either kept actively in the control loop at all times or is constantly made aware
of the feel of the present aircraft configuration through some auxiliary task which he can
practice on during critical phase of the flight.

REFERENCES

[1] J. A. Franklin, “ Dynamics, Control, and Flying Qualities of V/STOL Aircraft ,” AIAA Education Series ,
Reston, Virginia, USA, 2002.
[2] M. R. Napolitan o, “Aircraft Dynamics. From Modeling to Simulation, John Wiley & Sons, Inc.,USA,
2012.
[3] C. E. Moldoveanu, A. Giovannini, H. C. Boisson, Turbulence receptivity of longitudinal vortex –
dominated flows Lecture Notes in Computational Science and Engineering, Vol. 69: BAIL 2008 –
Boundary and Interior Layer Proceedings of the International Conference on Boundary and Interior
Layers – Computational and Asymptotic Methods, Limerick, Springer Verlag, Berlin, pag. 227 – 236,
ISBN: 978 -3-642-00604- 3, July, 2008.
[4] C. E. Moldoveanu, A. Giovannini, H. C. Boisson, P. Șomoiag , Simulation of Wake Vortex Aircraft in
Ground Effect, INCAS Buletin, Volum 3, nr. 1/2011, pag.55 -61, ISSN 2066 -8201, 2011.
[5] C. E. Moldoveanu, F. Moraru, P. Șomoiag, H. C. Boisson, A. Giovan nini, Turbulence Modelling
Applied to Aircraft Wake Vortex Study , MTA Review, Volum XXI, nr. 1/ 2011, ISSN 1843 -3391,
(2011).
[6] A. N. Tudosie, Supersonic Air Inlet Control System Based On The Inner Channel Minimum Cross –
Section Position Control, International Conference on Military Technologies ICMT 2009, Brno, Czech
Republic, 2009 .
[7] A. N. Tudosie, Aircraft Jet Engine Exhaust Nozzle Controller Based on Turbine Pressure Ratio Sensor
with Micro -Jet System , Proceedings of the 11th Inte rnational Conference on Applied and Theoretical
Electricity (ICATE), Craiova, 2012.
[8] C. Rotaru and I. Sprintu, “ State Variable Modeling of the Integrated Engine and Aircraft Dymanics ”,
10thInternational Conference on Mathematical Problems in Engineering , Aerospace and Sciences
(ICNPAA 2014)” , Volume 1637, pp. 889 -898, 2014.

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