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American Institute of Aeronautics and Astronautics

1

Exponential Mapping Controller Applied to Aircraft

Hildebrando F. de Castro1, Pedro Paglione2and Carlos Henrique Ribeiro3

Instituto Tecnolgico de Aeronutica, So Jos dos Campos, So Paulo, 12228-900, Brazil

A novel modified exponential function to achieve tracking and regulatory direct and

indirect model-free feedback control of a class of nonlinear dynamical systems is presented.

Its simplicity was a requisite to have it run on equipment and devices with memory and

processor constraints. The control algorithm needs only two parameters and can be applied

to systems based on knowledge about its free response and expected disturbances. Its main

advantages are relative ease of implementation and intuitive tuning. The controller is

applied to the simulation of two different aircraft under stochastic winds and wind shear

and its results are compared to a classical implementation. EMC presented promising

results. Its intuitive form of parameterization allowed in some cases for immediate good

results or at least for good initial estimatives for later tuning.

Nomenclature

u = forward speed

v = lateral speed

w = vertical speed

p = roll rate

q = pitch speed

r = yaw speed

x = position along thexaxis

y = position along theyaxis

H = altitude

= roll angle

= pitch angle

= heading

= speed = indicated air speed = angle of attack = sideslip angle = flight path angle (FPA) = heading rate = mass = mass change ratea = aileron position

e = elevator position

r = rudder position

= throttle position

= initial mass = wing area = wing span = mean aerodynamic chord1Doctoral Student, Departmento de Mecnica do Voo, Praa Mal. Eduardo Gomes, 50 Vila das Accias.2Professor, Departmento de Mecnica do Voo, Praa Mal. Eduardo Gomes, 50 Vila das Accias.3Professor, Departmento de Teoria da Computao, Praa Mal. Eduardo Gomes, 50 Vila das Accias.

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2

= moment of inertia about x axis = moment of inertia about y axis = moment of inertia about z axis = moment of inertia about xz plane

= maximum propulsive force

= velocity influence on propulsive force = density influence on propulsive force = reference speed = thrust reference speed = reference air density = specific fuel consumption = induced drag = induced drag constant = windxposition, world coordinates = windyposition, world coordinates = wind radius

= wind strength, radial direction

= wind strength, vertical direction = lift coefficient = drag coefficient = side force coefficient = rolling moment coefficient = pitching moment coefficient = yawing moment coefficient = reference drag coefficient = reference lift coefficient = aircraft lift curve slope

= lift due to pitch rate

= lift due to elevator = reference pitching moment = pitching moment due to pitch rate = pitching moment due to angle of attack rate = pitching moment due to elevator = side force due to sideslip = side force due to aileron = side force due to rudder = rolling moment due to sideslip

= rolling moment due to roll rate

= rolling moment due to yaw rate = rolling moment due to aileron = rolling moment due to rudder = yawing moment due to sideslip = yawing moment due to roll rate = yawing moment due to yaw rate = yawing moment due to aileron = yawing moment due to rudder

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I. Introduction

ontrol of a dynamical system can be based on its model or on different levels of prior knowledge about its free

response. Model-based linear techniques for control abound, but systems and disturbances are usually of a

nonlinear nature and can be extremely hard to model. Imprecisions may come from uncertainties about the plant andits external disturbances, or from a purposeful choice of model simplification1. This may lead to poor or inadequate

response in a model-based control project.

Classical (e.g.robust and adaptive control) and Artificial Intelligence-based (e.g.fuzzy logic control, neural and

neuro-fuzzy) techniques exist to deal with uncertainties in nonlinear systems. While classical techniques will rely on

a model or on a simplified model of the plant to be controlled, Artificial Intelligence (AI) techniques use heuristics

or other methods to learn or to identify the plant behavior, and then to execute control actions.

Implementation of a control system can be a time-consuming task if the system is unknown or poorly known.

The derivation of a control-oriented model can take up to 90% of a project global time, and requires much

knowledge of the system to be modelled and decisions about the physical model structure, parameter identification

and experimental validation2. Maintenance of controller parameters is also a big concern in the industry. Since more

than 90% of the installed and running controllers are of the Proportional-Integral-Derivative (PID) type 3, changes in

the process materials, equipment ageing and campaigns shifting demand maintenance personnel and engineering to

continuously monitor and reprogram Distributed Control Systems (DCSs), Programmable Logic Controllers (PLCs)and other devices to guarantee proper response of the process. Moreover, most current DCSs in use were installed in

the 1970s and 1980s, and their processing power and network capabilities are limited 4. While there is a trend to

update this hardware, this is a slow task because they are usually replaced only during maintenance stoppages.

With these aspects in mind, a model-free, low memory and processor consuming, intuitive to implement and

tune controller is then desirable. We thus propose here a novel sliding mode- and fuzzy logic-inspired exponential

mapping controller (EMC) which seeks to take advantage of both the conceptual simplicity of the bang-bang-based

Sliding Mode Control (SMC) and the lack of need of a model, shape-modifiable response and use of heuristics for

tuning of the Fuzzy Logic Controller (FLC).

The rest of the paper is organized as follows. Sections 2 and 3 review concepts and bibliography related to SMC

and FLC, respectively. Section 4 presents the EMC controller. Section 5 shows its implementation and simulation

results for two different aircraft. Finally, Section 6 presents the conclusions of the paper.

II.

Sliding Mode Control

Sliding Mode Control (SMC) is known as a method for dealing with nonlinear systems with uncertain dynamics

and disturbances because of its order reduction property and low sensitivity to disturbances and process parameter

variations, which relaxes the burden of the need for exact modeling5. It does so by application of a discontinuous

control signal that forces the system to slide along a specified surface of the systems normal behavior.

For second order systems, the sliding surface (a line) can be defined as:

(1) (2)

where is the system output and is the reference to be followed, is a constant, is the error and its firstderivative. The control law consists of two parts:

(3) () (4)where is the control input, is the equivalent control term, is the corrective term, is a strictly positive

constant and ()is the sign function. The equivalent control6uses the available model of the system dynamicsand the sliding surface definition to determine a control action that would keep the state on the sliding surface, if the

model were perfect. The corrective term guarantees that the state will be drawn to and remain on the sliding surface

C

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despite model and disturbances uncertainties. However, this term can also be a cause of chattering, which occurs in

systems with delays in their actuators or limited sampling rate and can lead to high oscillatory control efforts and

ultimately instability. Slotine and Li1proposed a way to smooth it out by means of a boundary layer neighboring the

so called switching surface. This can be accomplished by replacing the sign function by a saturation function such

that:

(5)where:

() { || (6)and is the term that defines when interpolation will occur. Song and Smith7observed that both and can bederived using open loop experimental data. They proposed to apply maximal input and record the saturation result to

define . Additionally, the disturbance magnitude can be used to define . Standard linear controller design can onlylinearly approximate this nonlinear curve.

A hyperbolic tangent function can also be used:

(7)Finally, a bang-bang-type SMC is a direct switching control strategy that drops the equivalent term:

() (8)where is large enough to suppress all bounded uncertainties and unstructured systems dynamics 8. The

Fillipov method9is used for its design, by which a sufficient large local attractor is defined for the system trajectory.

III.

Fuzzy and Neuro-Fuzzy ControlFuzzy logic, proposed by Zadeh10 in his seminal work for logic calculus and later extended11 is a way to

approach problems that are difficult or unnatural to define in a binary, crisp sense.

Fuzzy systems use fuzzy sets. A fuzzy set is a set without a clearly defined boundary. Its elements contain only a

partial degree of membership:

() [] (9)where ()is the membership function for a fuzzy set , is an element of the set and is the universe of

discourse. It means that can have degrees of membership from 0 to 1, inclusive. Thats much different fromordinary logic that accepts only true or false.

Fuzzy sets are subject to set operations such as union, intersection, and complement, which express logic

statements or propositions. Fuzzy relations can be represented linguistically by natural language statements in the

form of fuzzy IF or antecedents, and THEN or consequent rules. Table 1 below shows an example of a set of fuzzyrules for a speed control system.

Table 1Fuzzy rules for a speed control system

IF THEN

Speed is too slow Accelerate hard

Speed is slow Accelerate

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Speed is ok Do nothing

Speed is fast Break

Speed is too fast Break hard

The first implementation of fuzzy logic to dynamic control was performed by Mamdani 12. Fuzzy Control is

currently used in many industrial applications13and in many other fields14,15.The process of using fuzzy logic to implement a control system consists of a fuzzification of the input variables

( and , for example, for a SISO control), application of the fuzzy operators, matching of the antecedents andinference of consequent parts, aggregation of the consequents and finally, defuzzification16. The fuzzy aspect of the

rules tackels the imprecise definition of the system, allowing for imprecisions in the design of the control system to

be tolerated to a certain degree and eliminating the need for a well-defined mathematical model of the plant17, but

the translation of these linguistic rules into actions depend on the choice of certain parameters, for which there are

no formal methods to get. These parameters, however, can be defined by adaptive schemes, such as those defined by

neural network architectures, designed to learn from training data, but which are in general not able to profit from

structural knowledge18.

A neural network (NN) is a mathematical structure comprised of neurons and layers. A three-layer feed-forward

NN is capable of approximating any nonlinear function19. The fundamental building block for a NN is the neuron,

consisting of a weight function, a net input function and the transfer function:

( ) (10)where is the input, is the weight, is a constant bias, ()is the transfer function, usually linear for the

output layer and nonlinear for the internal ones and is the output. The parameters and are both adjustablescalars. This flexibility allows for optimization algorithms to be employed in order for convergence or learning to

occur.

A. The Neo-Fuzzy NeuronThe Neo-Fuzzy Neuron (NFN) is a type of neuro-fuzzy structure that allows fast training time and relatively low

processing demands20. The input signals through the nonlinear transfer function are calculated as:

()

(11)

where is the number of inputs. For each nonlinear transfer function, the following fuzzy inference with asingleton consequent is computed:

(12)where is a fuzzy set whose membership function is and is a singleton.Since each membership function in the antecedent is triangular and complementary, an input signal activates

only two neighboring membership functions simultaneously so the sum of the grades of these membership functions

is always 1. Thus, the defuzzification taking a center of gravity does not need a division:

() () () (13)The NFN then uses gradient descent to reduce the errors through the adjustment of the weights . In the workof Gouva21 this already relatively simple structure is further minimized in the so called ONFC (Online Neurofuzzy

Controller). The ONFC uses only one input, two linear membership functions and one output for SISO feedback

control, so only two gains have to be adjusted. This simplicity allows it to be implemented in a DCS function block.

Carvalho22 implemented a modified ONFC in a refinery coke processing unit with very good results.

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IV.

The EMC

The proposed EMC (Exponential Mapping Controller) seeks to make use of some of the advantages of SMC and

FLC in a simplified and intuitive way. The relative ease of implementation of a bang-bang SMC coupled with a

boundary layer to deal with chattering and the lack of need of a prior model of the system and intuitiveness of

implementation of a FLC.

Yagiz and Hacioglu23 proposed improving the response within the boundary layer in a SMC through

dynamically deciding the slope of a linear sliding surface. Their results are compared to a traditional constant slopelinear sliding surface SMC with good results. Ha et al.24presented a way to tune a fuzzy SMC (FSMC) by rotating

and shifting a nonlinear sliding surface and achieved faster reaching times and improved tracking error. Tokat et

al.25proposed a parabolic function coupled to a shifted sigmoid function to change the linear discontinuous control,

the equivalent term was computed with a simplifed yet model-based piecewise function. Choi and Park26proposed a

Moving Sliding Surface (MSS) that was dependent on the initial conditions of the system and through rotations and

shifts would subsequentely move it to a predetermined switching surface. Park and Choi 27generalized the MSS to

higher-order SMCs by checking the conditions to rotate and shift the sliding surface. These approaches, however,

make use of the equivalent control and therefore need a model of the system. Tabatabaei et al. 28 proposed a

sigmoidal mapping function for control of molten metal pouring, but it needs eleven parameters to be defined and

ten rules for the adaptation of the algorithm.

EMC implements a bang-bang SMC-inspired approach with a heuristically-defined nonlinear mapping function

comparable to a bounded linear sliding surface. The shape and boundary of the mapping function are defined by the

plant operator based on partial knowledge about the system to be controlled and about expected disturbances. Theboundary is derived from basic information about the open loop response of the plant and its shape is defined based

on knowledge about the system behavior.

The number and format of the membership functions in an FLC allow for altering the shape of its response

function. This can be simplified by using only one shape-modifiable function. This in turn accounts for easier

implementation since there is no need to define the number and shape of the membership functions and allows for

faster computation times and more modest memory requirements, at the cost of a limited set of shapes.

Since in practice EMC needs only two parameters its implementation and tuning are much simplified. This

simplicity allows for restoring the responsability of tuning back to the systems operator or engineer. PIDs have three

or more non-intuitive parameters to be tuned. As a matter of fact, the number of parameters to tune an industrial

scale PID controller is usually five. Typical SISO FLCs, even though intuitive to implement, typically need at least

nine parameters to be tuned. We understand, however, that tuning only a small number of intuitive parameters is

crucial for the implementation of automatic control systems in the industry. An alternative would be designing

adaptable, untunable and hands-off controllers, but those depend on a high level of confidence - usually derived

from design efforts with long duration and expensive costs - before its implementation can be carried out. Also,controllers maintenance is another key issue for the industry which further motivate the design and use of relatively

easy-to-tune controllers.

EMC is implemented as follows. First it calculates the error, as in equation 1, redefined here:

(14)Then a switching line is computed:

(15)where

is heuristically defined as being the acceptable error before applying full control input. For lower order

or non-oscillatory systems the term can usually be dropped. Then is restricted to vary between -1 and 1 to bewithin the exponential function allowable range: { (16)

And then an exponential function is computed:

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() || (17)where is the parameter that alters the shape of the exponential function, allowing for a class of

nonlinearities and disturbances to be dealt with.

Finally, the controller output is calculated:

( ) (18)where and are the actuator minimum and maximum positions, respectively.EMC can be used for direct and indirect control. The above equations are for the direct case. Direct control is

used when for , otherwise indirect control must be used. For the indirect case equation (18) becomesequation (19):

( ) (19)where and are the controller inputs in time and , respectively and and are the

actuator minimal and maximal changes during the sampling period .Figure 1 presents the error

and controller output

for the EMC direct exponential function, for

,

and , for , not considering .

Figure 1. Plot of and , .Figure 2 emphasizes the difference in response due to .

-15 -10 -5 0 5 10 15

-10

-5

0

5

10

e

uEMC

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Figure 2. Plot of

and

,

(solid),

(dashed) and

(dotted).

For systems that need a high control effort even with small errors, should be defined in the range , or otherwise. Systems with increasing levels of disturbance should alleviate the actuatorsresponse by decreasing .

One issue that arises with this heuristic approach is for systems where instability happens before saturation. In

this case, knowledge about the system behavior before instability must be used. One example is the control of

aircraft altitude with the elevator as the sole actuator. In this case when full elevator is applied continuously, the

aircraft will eventually stall or even spin so the ratio between full elevator and altitude gain cannot be determined

directly. Song and Smith29used the same approach for heading control of an AUV using only the vertical rudder.

With application of full rudder, the AUV will eventually reach a maximal turn speed. The resulting function is the

sliding, or switching, line.

V.

Aircraft ControlTwo aircraft, very different dynamically, were chosen for implementation of the EMC. Autopilots were built for

a widebody jet airliner and for a small single-engine aircraft. The reader is referred to Stevens and Lewis 30 for a

thorough discussion of aircraft control.

The vector for the states of the aircraft is shown in (20). The control variables are shown in (21).

[ ] (20)[ ] (21)

Two different flight control systems were implemented. One controls ascent and descent and coupledwith a Stability Augmentation System (SAS) and the other is a cruise autopilot with speed hold, altitude hold and

roll angle hold also coupled with an SAS. The SAS for both control systems stabilize

,

,

and

. They will be

referenced as CAP, for Cruise Autopilot and IACAP, for Constant Indicated Airspeed Climb and Descent Autopilot,respectively. Both flight control systems were implemented using a Linear Quadratic Tracker with Output

Feedback30(LQTOF) and EMC.

The controllers were implemented using MATLABand Simulink, an environment to perform generic matrix-

based mathematical calculations and simulations. The aircraft were initially trimmed to a specific flight condition.

For the CAP trimming was performed to maintain constant , , , and relative position of the center ofgravity (C.G.) and solving for , , , , , and .

For the IACAP trimming was performed to maintain , , and and solving for , , , , , and .

-15 -10 -5 0 5 10 15

-10

-5

0

5

10

uEMC

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After trimming, disturbances were applied to check the performance of the control systems. Stochastic winds

were generated using the continuous Dryden wind turbulance model, available in the Aerospace Blockset in the

MATLABenvironment and a wind shear was generated using a sinusoidal turbulence function for the horizontal

aspect and a semi-sinusoidal turbulence function for the vertical aspect. The wind shear model uses five parameters,

namely , , , and . All aircraft were simulated with both disturbances and all responses weresatisfactory but for the sake of brevity not all results are shown. Fuel consumption is considered in the models.

The main characteristics and coefficients for the simulated aircraft, including mass, geometry, inertia andpropulsion properties are given in the Appendix. The adimensional force and moments coefficients are calculated

according to equations (22) to (27).

(22) (23)

(24) ( ) (25) ( ) (26) ( ) (27)

The propulsive force is calculated according to equation (28).

(28)Finally, the loss of mass due to fuel consumption is calculated according to equation (29).

(29)

A.

The Widebody Jet Airliner

A widebody jet airliner (comparable to an Airbus A310) was used to simulate a cruise autopilot. The actuators

for the jet are modeled using first order transfer functions, saturation and rate limiters. Their values and

configuration are available in the Appendix.

B.

Widebody Jet Airliner LQTOF CAP

To hold and a longitudinal compensator was coupled with LQR gains and control inputs to and . Thestate-space compensator is shown in (30). The LQR gains are shown in (31).

(30) (31)

A PID controller was used to control through . Its parameters were: (32)

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The states , , and were stabilized in the SAS inner loop using the LQR gains shown in (33) through ,and .

(33)

Figure 3 presents a diagram of the control architecture. is a vector containing the reference states, which areconstant for the CAP but become the reference trajectory for the IACAP.

Figure 3. LQTOF Control Architecture

Results for the LQTOF controller are shown in Fig. 4, with Dryden wind and wind shear. The equilibrium points

for the simulation were , , and . The parameters for the wind shear were , , , and , meaning that the aircraft reaches thewind at approximately 10 seconds of flight.

a)

0 20 40 60 80228.5

229

229.5

230

230.5Speed

t [s]

V

[m/s]

0 20 40 60 807990

7995

8000

8005Altitude

t [s]

H[m]

0 20 40 60 80-10

-5

0

5

10Roll Angle

t [s]

[

]

0 20 40 60 80-6

-4

-2

0

2Sideslip

t [s]

[

]

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b)

Figure 4. LQTOF responses for the widebody jet, Dryden wind and wind shear. a) States. b) Actuators.

Results for the LQTOF controller with wind shear only are shown in Fig. 5.

a)

0 20 40 60 800

50

100Throttle

t [s]

[%]

0 20 40 60 80-4

-2

0

2Elevator

t [s]

p

[]

0 20 40 60 80-10

-5

0

5Aileron

t [s]

a

[]

0 20 40 60 80-4

-2

0

2

4Rudder

t [s]

r

[]

0 20 40 60 80228

229

230

231Speed

t [s]

V[m/s]

0 20 40 60 807990

7995

8000

8005Altitude

t [s]

H[m]

0 20 40 60 80-10

-5

0

5

10 Roll Angle

t [s]

[

]

0 20 40 60 80-6

-4

-2

0

2 Sideslip

t [s]

[

]

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b)

Figure 5. LQTOF responses and actuators for the widebody jet, wind shear only. a) States. b) Actuators.

C. Widebody Jet Airliner EMC CAP

As in the classical controller, was controlled through . , and were controlled through , and were controlled through and was controlled through . The corresponding signals were added for eachindividual actuator.

Figure 6 presents the control architecture for the EMC CAP and the EMC array of controllers, where is theerror vector for each state and is the controller output vector, containing , , and , respectively. The ErrorCalculation block computes the errors from the equilibrium states.

a)

0 20 40 60 800

50

100Throttle

t [s]

[%]

0 20 40 60 80-4

-2

0

2Elevator

t [s]

p

[]

0 20 40 60 80-10

-5

0

5Aileron

t [s]

a

[]

0 20 40 60 80-4

-2

0

2

4Rudder

t [s]

r

[]

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b)

Figure 6. Simulink model for the EMC CAP. a) Control architecture. b) EMC array of controllers.

The parameters for the EMC CAP implementation are shown in Table 2.

Table 2EMC parameters for the CAP

State -2.5m/s -1 -25 +25 35m 1 -25 +25 25 0 -25 +25 8 0 -25 +25 45/s -0.5 -25 +25 -30/s 0.25 -25 +25 -150/s 1 -30 +30

Results for the EMC controller are shown in Fig. 7, with Dryden wind and wind shear. The equilibrium points

and wind shear parameters are the same as for the LQTOF controller.

V

H

1

u

e u

EMC_theta

e u

EMC_r

e u

EMC_q

e u

EMC_phi

e u

EMC_p

e u

EMC_V

e u

EMC_H

1

e

phi

theta

q

p

r

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a)

b)

Figure 7. EMC responses for the widebody jet, Dryden wind and wind shear. a) States. b) Actuators.

Results for the EMC controller with wind shear only are shown in Fig. 8.

0 20 40 60 80229.8

229.9

230

230.1

230.2Speed

t [s]

V[m/s]

0 20 40 60 807990

7995

8000

8005Altitude

t [s]

H[m]

0 20 40 60 80-5

0

5Roll Angle

t [s]

[

]

0 20 40 60 80-2

-1

0

1

2Sideslip

t [s]

[

]

0 20 40 60 8065

70

75Throttle

t [s]

[%]

0 20 40 60 80-1.5

-1

-0.5

0Elevator

t [s]

p

[]

0 20 40 60 80-10

-5

0

5

10Aileron

t [s]

a

[]

0 20 40 60 80-5

0

5

10x 10

-3 Rudder

t [s]

r

[]

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a)

b)

Figure 8. EMC responses for the widebody jet, wind shear only. a) States. b) Actuators.

D.

Widebody Jet Airliner LQTOF IACAP

To hold during a climb or descent maneuver a longitudinal compensator was coupled with LQR gains andcontrol inputs to

. The state-space compensator is shown in (34). The LQR gains are shown in (35).

[] [] (34) [ ] (35)

A PID controller was used to control through . Its parameters were the same as for the CAP.States , , and were stabilized in the SAS inner loop using the LQR gains shown in (36) through ,and.

0 20 40 60 80229.9

229.95

230Speed

t [s]

V[m/s]

0 20 40 60 807990

7995

8000

8005Altitude

t [s]

H[m]

0 20 40 60 80-4

-2

0

2

4Roll Angle

t [s]

[

]

0 20 40 60 80-2

-1

0

1Sideslip

t [s]

[

]

0 20 40 60 8067

68

69

70

71Throttle

t [s]

[%]

0 20 40 60 80-1

-0.9

-0.8

-0.7Elevator

t [s]

p

[]

0 20 40 60 80-5

0

5Aileron

t [s]

a[

]

0 20 40 60 80-2

0

2

4x 10

-3 Rudder

t [s]

r[

]

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(36)

Results for the LQTOF controller are shown in Fig. 9, with wind shear only. The equilibrium points for the

simulation were

,

,

,

and

. The parameters for the wind shear

were , , , and , meaning that the aircraft reachesthe wind at approximately 23 seconds of flight.

Figure 9. LQTOF responses for the IACAP for the Widebody Jet Airliner, wind shear only. a) States. b)

Actuators.

0 20 40 600

500

1000Altitude

t [s]

H[m/s]

0 20 40 60120

140

160

180Indicated Airspeed

t [s]

VIAS

[]

0 20 40 60-40

-20

0

20

40Flight Path Angle

t [s]

[m]

0 20 40 60-5

0

5

10

15Roll Angle

t [s]

[

]

0 10 20 30 40 50 60 70-50

0

50Elevator

t [s]

e

[]

0 10 20 30 40 50 60 70-5

0

5Aileron

t [s]

a

[]

0 10 20 30 40 50 60 70-5

0

5Rudder

t [s]

r

[]

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E.

Widebody Jet Airliner EMC IACAP

As in the classical controller, was controlled through . , and were controlled through , and were controlled through and was controlled through . The corresponding signals were added for eachindividual actuator. In this case, the reference condition changes with time.

The IACAP EMC parameters are shown in Table 3.

Table 3EMC parameters for the IACAP

State 8m/s 3.5 -25 +25 2.5m -0.75 -25 +25 -25 0 -25 +25 15 2 -25 +25 20/s 3 -25 +25 30/s 3 -25 +25 10/s 4 -30 +30

Results for the EMC controller are shown in Fig. 10, with wind shear only. The equilibrium points and wind

shear parameters are the same as for the LQR controller.

a)

0 20 40 600

500

1000Altitude

t [s]

H[m/s]

0 20 40 60140

150

160

170


t [s]

VIAS

[]

0 20 40 60-5

0

5

10

15Flight Path Angle

t [s]

[m]

0 20 40 60-2

-1

0

1

2Roll Angle

t [s]

[

]

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b)

Figure 10. EMC responses for the widebody jet, wind shear only. a) States. b) Actuators.

F. The Small Single-Engine Aircraft

A small single-engine aircraft (comparable to a North American Aviation Navion) was used to simulate a cruise

autopilot. The actuators for the aircraft are modeled using first order transfer functions, saturation and rate limiters.

Their values and configuration are available in the Appendix.

G. Small Single-Engine Aircraft EMC CAP

EMC was used to implement the CAP for a small single-engine aircraft. Its parameters (previously shown in

Table 2) and control architecture are the same as for the widebody jet airliner. The equilibrium points for the

simulation were

,

,

and

. The parameters for the wind shear were

, , , and , meaning that the aircraft reaches the wind atapproximately 43 seconds of flight.It is worth noting that even though the dynamics of a widebody jet airliner and a small single-engine aircraft aremuch different, the EMC parameters did not have to be changed. This presents some robustness that is still under

analysis.

Results for the EMC CAP controller are shown in Fig. 11, with Dryden wind and wind shear.

0 10 20 30 40 50 60-20

0

20Elevator

t [s]

e

[]

0 10 20 30 40 50 60-10

0

10 Aileron

t [s]

a

[]

0 10 20 30 40 50 60-10

0

10Rudder

t [s]

r

[]

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a)

b)

Figure 11. EMC responses for the single-engine aircraft, Dryden, wind and wind shear. a) States. b)

Actuators.

Results for the EMC controller with wind shear only are shown in Fig. 12.

0 50 10057

58

59

60

61Speed

t [s]

V[m/s]

0 50 100960

980

1000

1020Altitude

t [s]

H[m]

0 50 100-1

-0.5

0

0.5

1Roll Angle

t [s]

[

]

0 50 100-4

-2

0

2Sideslip

t [s]

[

]

0 50 10060

70

80

90

100Throttle

t [s]

[%]

0 50 1000

1

2

3Elevator

t [s]

p

[]

0 50 100-4

-2

0

2

4Aileron

t [s]

a[]

0 50 100-5

0

5Rudder

t [s]

r

[]

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a)

b)

Figure 12. EMC responses for the single-engine aircraft, wind shear only. a) States. b) Actuators.

H. Small Single-Engine Aircraft EMC IACAP

The EMC IACAP implementation uses the same parameters shown in Table 3. The architecture is the same as

previously shown.

Results for the EMC controllers are shown in Fig. 13, with Dryden wind and wind shear. The equilibrium points

for the simulation were , , , and . The parameters for the windshear were , , , and , meaning that the aircraftreaches the wind at approximately 194 seconds of flight.

0 50 10057

58

59

60Speed

t [s]

V[m/s]

0 50 100960

980

1000

1020Altitude

t [s]

H[m]

0 50 100-0.2

0

0.2

0.4

0.6Roll Angle

t [s]

[

]

0 50 100-4

-2

0

2Sideslip

t [s]

[

]

0 50 10070

80

90

100Throttle

t [s]

[%]

0 50 1001

1.5

2

2.5Elevator

t [s]

p

[]

0 50 100-1

-0.5

0

0.5

1Aileron

t [s]

a[]

0 50 100-4

-2

0

2

4Rudder

t [s]

r

[]

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a)

b)

Figure 13. EMC responses for the single-engine aircraft, Dryden wind and wind shear. a) States. b)

Actuators.

VI.

Conclusion

An exponential-function based controller inspired by the bang-bang SMC and NFN was proposed. It seeks to

ease the task of implementing and maintaining a nonlinear control system. EMC exhibits the property of allowing its

tuning using only two parameters. The first () is the most important one, since during the simulations it was seenthat its value can be critical in terms of stability, but a heuristic helps estimating its initial values. The secondparameter () allows for specifying a smoother or a more aggressive response. Some of the implemented EMCsshowed good results with the first estimate of their parameters, obtained from the simulation. The classical

controllers required the model of each aircraft and each controller system used 43 parameters. EMC managed to

achieved roughly the same results without any analytical information about the aircraft and used 14 heuristically-

defined parameters. The authors relied only on the aircraft open loop response and behavior. Using Lyapunov to

check its stability properties, robustness characteristics and parameter range validation are intended for further

0 100 200 3000

500

1000Altitude

t [s]

H[m/s]

0 100 200 30046

48

50

52


t [s]

VIAS

[]

0 100 200 300-0.2

0

0.2

0.4

0.6Flight Path Angle

t [s]

[m]

0 100 200 300-1

0

1

2Roll Angle

t [s]

[

]

0 50 100 150 200 250 300-50

0

50Elevator

t [s]

p

[]

0 50 100 150 200 250 300-10

0

10Aileron

t [s]

a

[]

0 50 100 150 200 250 300-5

0

5Rudder

t [s]

r[

]

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development of the proposed controller. The EMC has already been implemented in a real laboratory system,

namely the Quanser 3D Helicopter, with good results.

Appendix

Aircraft Main Characteristics

Parameter Widebody Jet

Airliner

Single-engine

Aircraft

Length 46.66m 8.3m

Height 15.80m 2.6m

Maximum Takeoff

Weight

157,000kg 1,247kg

Cruise Speed 861km/h 250km/h

Aircraft Geometry, Mass, Inertia and Propulsion Properties

Variable Widebody Jet Single-engine 120,000kg 1,123.7kg 260m 17.09m 43.89m 10.18m 6.61m 1.74m

5.55e6kg/m2 1415.4kg/m2 9.72e6kg/m2 3999.7kg/m2 14.51e6kg/m2 4756.7kg/m2 -3.3e4kg/m2 -142.4kg/m2

240,000N 2,200N 0 -1 0.75 0.75

240m/s 45m/s

1.225kg/m 1.225kg/m 1.8e-5 6e-6 0.06 0.075 0 -0.045

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Widebody Jet Airliner and Small Single-Engine Aircraft Actuators

Actuator First Order Dynamics Saturation Limits Rising and Falling Rate Limits

[] [] [] [] [] [] [] []

Aircraft Coefficients

Variable Widebody Jet Single-engine

0.0175 0.0468 0 0 4.982 4.44 -1.4 3.8 0.435 0.353 -0.025 0 -30 -4.98 -10 -2.18 -1.46 -0.923 -1.5 -0.564 0.05 0 0.3 0.157 1.3 -0.074 -3.9157 -1.2012 0.8735 0.3135 -0.33 -0.134 0.25 0.107 1.75 0.071 -0.4518 -0.1685

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-2.2591 -0.3662 -0.125 -0.0035 -1 -0.072

Actuator Dynamics Simulation Block

The reference values for the actuators are constant for the CAP. For the IACAP they are constantly updated,

except for the throttle, which is constant.

Aircraft and Wind Simulation Block

References

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