aircraft landing control design based on artificial life...

Indian Journal of Engineering & Materials Sciences Vol. 20, June 2013, pp. 165-176

Aircraft landing control design based on artificial life and CMACs

Jih-Gau Juang*, Cheng-Yen Yu & Chung-Ju Cheng

Department of Communications, Navigation and Control Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan

Received 29 July 2012; accepted 21 February 2013

This paper presents the use of different artificial life-based optimization algorithms and cerebellar model articulation controllers (CMACs) in aircraft automatic landing control. The proposed intelligent control system can act as an experienced pilot and guide the aircraft landed safely in wind disturbance condition. Lyapunov theory is applied to obtain adaptive learning rule and stability analysis is also provided. The proposed controllers have better performance than conventional controller.

Keywords: CMAC, Landing control, Optimization algorithms, Wind disturbance.

According to International Air Transport

Association (IATA) classification, aircraft accident contributing factor is divided into five categories as human factor, technical, environment, organization factor and insufficient. An aircraft accident survey of 19562 aircraft accidents from 1994 through 2003, weather was a contributing factor. The percentage of weather related to total accidents is 21.3%1. The first automatic landing system (ALS) was developed in England in 1965. Since then, most aircraft have had this system installed. The ALS relies on the instrument landing system (ILS) to guide the aircraft into the proper altitude, position, and approach angle during the landing phase. According to Federal Aviation Administration (FAA) regulations2, environmental conditions considered in the determination of dispersion limits are: headwinds up to 25 knots, tailwinds up to 10 knots, crosswinds up to 15 knots, moderate turbulence, and wind shear of 8 knots per 100 feet from 200 feet to touchdown. Conventional automatic landing systems can provide a smooth landing, which is essential to the comfort of passengers.

However, these systems work only within a specified operational safety envelope. When the conditions, such as turbulence or wind shear, are beyond the envelope, they often cannot be used. Most conventional control laws generated by the ALS are based on the gain scheduling method3. Control parameters are preset for different flight conditions

within a specified safety envelope, which is relative defined by FAA regulation. When the flight conditions are beyond the envelope, the ALS is disabled and manual operation is engaged. An inexperience pilot may not be able to guide the aircraft safely. Therefore it is desirable to develop an intelligent ALS that expands the operational envelope to include more safe responses under a wider range of disturbance conditions. The goal of this study is to show that the proposed intelligent ALS can relieve human operation and guide the aircraft to a safe landing in turbulence environment.

PID control has been applied to controller design for decades. It is the most used controller in engineering applications. Control gains of the PID can be tuned by many techniques. In recent years, genetic algorithm (GA) is the most used one. This study applies Artificial Life models to modeling and simulating life-like phenomena for aircraft automatic landing controller design. This paper is mainly based on the following four algorithms to adjust the control parameters of the autopilot, they are Bacterial Foraging Optimization (BFO)4, Particle Swarm Optimization (PSO)5, Chaos Particle Swarm Optimization (CPSO)6 and Bacterial Swarm Optimization (BSO)7. Control schemes are based on the Cerebellar Model Articulation Controller (CMAC)8.

In recent years, intelligent control is popular in control engineering applications. Many intelligent concepts, such as fuzzy system and neural network9,10, have been applied into various scientific and

__________ *Corresponding author (E-mail: [email protected])

INDIAN J. ENG. MATER. SCI., JUNE 2013

166

engineering researches. There are also obvious achievements in flight control domain11-16. CMAC has attracted researchers’ attentions for the following advantages: rapid learning speed, guarantee of learning convergence, good generalization, and low computational load. A simplified addressing technique procedure, S_CMAC, was presented by Chiang et al.17. The simplified model not only provides fixed addressing procedure for CMAC but also reduces the usage of memory. The S_CMAC still posses the same learning convergence characteristics of CMAC, but with more enhanced accurate system performance. The calculation speed of S_CMAC is faster than conventional CMAC. It is very useful in on-line control system. But it is hard to decide the number of layers and the value of learning rate of the S_CMAC. In this study, we utilize a self-organizing CMAC (SOCM)18 by judging error rate to add or delete layer and introduce a SOCM with adaptive learning rate to automatic landing system. Because adaptive control theory has been successfully employed in various fields, more and more researchers integrated the adaptive concept into the intelligent control system. The performance of the intelligent ALS under severe disturbance can be improved by the advantages of the SOCM with adaptive learning rate, and the advantages are local generalization, rapid learning convergence, and environment adaptive capability.

System Description At the aircraft landing phase, the pilot descends

from the cruise altitude to an altitude of approximately 1200 ft above the ground. The pilot then positions the aircraft so that the aircraft is on a heading towards the runway centerline. When the aircraft approaches the outer airport marker, which is about 4 nautical miles from the runway, the glide path signal is intercepted, as shown in Fig. 119. As the airplane descends along the glide path, its pitch, attitude, and speed must be controlled. The descent rate is about 10 ft/s and the pitch angle is between -5° to +5°. Finally, as the airplane descends 20 to 70 ft above the ground, the glide path control system is disengaged and a flare maneuver is executed. The vertical descent rate is decreased to 2 ft/s so that the landing gear may be able to dissipate the energy of the impact at landing. The pitch angle of the airplane is then adjusted, between 0° to 5° for most aircraft, which allows a soft touchdown on the runway surface.

A simplified model of a commercial aircraft that moves only in the longitudinal and vertical plane is used in the simulations for implementation ease14. The motion equations of the aircraft are given as:

0

( ) ( )

( )cos( )180

u g w g q

E E T T

u X u u X w w X q

g Z Z

… (1)

TTEE

qgwgu

ZZg

qUZwwZuuZw

)sin()180

(

)180

()()(

0

0 … (2)

TTEE

qgwgu

MM

qMwwMuuMq

)()( … (3)

q … (4)

0180

Uwh … (5)

where u is the aircraft longitudinal velocity (ft/s), w is the aircraft vertical velocity (ft/s), q is the pitch rate (rate/s), θ is the pitch angle (deg), h is the aircraft altitude (ft), δE is the incremental elevator angle (°), δT is the throttle setting (ft/s), γo is the flight path angle (-3°), and g is the gravity (32.2 ft/s2). The parameters Xi, Zi and Mi are the stability and control derivatives.

To make the ALS more intelligent, reliable wind profiles are necessary. There are several turbulence models20. The most used one is the Dryden form. The model is given by:

u

uugcg as

a

tNuu

21)1,0(

… (6)

2)(

)(31)1,0(

u

wwwg as

bsa

tNw

… (7)

Fig. 1–Glide path and flare path

JUANG et al.: AIRCRAFT LANDING CONTROL DESIGN

167

where ])51ln(

)510/ln(1[510

huu windgc ,

u

ou L

Ua , hLw ,

u

ou L

Ua ,

w

ow L

Ua ,

3w

ow

L

Ub , 3/1100hLu for

230h , 600uL for 230h , gcu u2.0 for

500h , hugcw *00098.05.02.0 for 5000 h .

The parameters are: ug is the horizontal wind velocity (ft/s), wg is the vertical wind velocity (ft/s), u0 is the nominal aircraft speed (ft/s), uwind510 is the wind speed at 510 ft altitude, Lu and Lw are scale lengths (ft), σu and σw are RMS values of turbulence velocity (ft/s), Δt is the simulation time step (s), N(0,1) is the Gaussian white noise with zero mean and unity standards deviation, ugc is the constant component of ug, and h is the aircraft altitude (ft). Figure 2 shows a turbulence profile with a wind speed of 30 ft/s at 510 ft altitude. Control Scheme

A simplified PID controller is applied to aircraft landing control. Its inputs consist of altitude and altitude rate commands along with aircraft altitude and altitude rate. Via aircraft landing controller we can obtain the pitch command, as shown in Fig. 3. Then, the pitch autopilot is controlled by pitch command. The pitch autopilot14 is shown in Fig. 4. In order to enable aircraft to land more steady when an aircraft approaches to the flare path, a constant pitch angle is added to the controller. In general, the PID controller is simple and effective but there are some drawbacks such as apparent overshoot and sensitive to external noise and disturbance. When severe turbulence is encountered the PID controller may not

be able to guide the aircraft to land safely. With CMAC compensator the proposed controller can overcome these disadvantages. The control scheme utilizes a conventional PID controller to stabilize the controlled system and train the CMAC to provide precise control. The gains of PID controller are adjusted based on experiences, what it provides are tolerable solutions, not desired solutions. The CMAC can effectively meliorate these disadvantages.

The output of CMAC is the sum of several values of hypercube and the input vector is the values of the aircraft status. The overall control scheme is described in Fig. 5, in which the control signal U is the sum of the PID controller output and the CMAC output. The inputs of the CMAC and PID controller are: altitude, altitude command, altitude rate and altitude rate command. The PID controller provides tolerable solutions. In each time step, the CMAC involves a recall process and a learning process. In the recall process, it uses the desired system output of the next time step and the actual system output as the address to generate the control signal UCMAC. In the learning process, the control signal of the pitch autopilot, U, is treated as a desired output. It is used to modify the weights of CMAC stored at desired location which is addressed by the actual system

Fig. 2–Turbulence profile

Fig. 3–PID-controller

Fig. 4–Pitch autopilot


168

output and the system output of the next time step. The output of the CMAC is the compensation for pitch command. When the wind shear is too strong, the ALS cannot control the aircraft to land safely. Here, we use a CMAC control scheme to improve the ability of wind shear resistance of the ALS.

In the general basis function CMAC (CMAC_GBF), the content of hypercube can be expressed as

wi (xs)= vibi (xs) … (8)

where bi (xs) is a general basis function and vi is a weight to be obtained through learning. The output of the CMAC_GBF can be written as.

N

isiiiss

Tss xbvaxwxy )](.[)()( ,a … (9)

where

].........[ ,2,1, NsssTs aaaa

].........[ 21 NT wwwW

which are a basis function selection vector (or so-called association cells) and the vector of memory contents of CMAC_GBF, respectively. The Gaussian functions are employed as the basis functions

,1

( ) ( )N

i s i j s jj

b x x

… (10)

with

])(

exp[)(2

2,

,

ij

ijjsjsij

mxx

… (11)

mij is the mean, σij is the variance and N is the number of variables in the target function. Consequently, the weight function is

)()( ,1

js

N

jijisi xvxw

… (12)

The output from the CMAC with Gaussian basis functions can be mathematically expressed as

2)ˆ(2

1ss yyE … (13)

The updated amount for iv can be set to

)())(ˆ(

)(

)(

)(

)(

, siissTss

e

v

i

si

si

s

s

s

se

v

ie

vi

xbaxwyN

v

xw

xw

xy

xy

e

e

E

N

v

E

Nv

a

… (14)

where v is the learning rate for v. The mean and variables of the Gaussian function can also be adjusted to increase the approximation capability. The updating rules for these parameters can be derived as

ije

mij m

E

Nm

2

,,

)(2)())(ˆ(

)(

)(

)(

)(

ij

ijissiiiss

Tss

e

m

ij

ij

ij

si

si

s

s

s

se

m

mxxbvaxwy

N

m

xw

xw

xy

xy

e

e

E

N

a

… (15)

))(2

)(())(ˆ(

)(

)(

)(

)(

3

2,

,ij

ijissiiiss

Tss

e

ij

ij

ij

si

si

s

s

s

se

ijeij

mxxbvaxwy

N

xw

xw

xy

xy

e

e

E

N

E

N

a

… (16)

where αm and ασ are the learning rates for mean and variance.

The learning method for S_CMAC_GBF and CMAC_GBF are the same17, but they are different in the structures. The quantified method of input variables is different from the quantified conventional CMAC_GBF. In S_CMAC_GBF, each layer has only one block, and each block has different basis function. The weighting of each layer can be obtained by multiplying basis function of each layer with its height. In S_CMAC_GBF, the output equation is the same as CMAC_GBF. as,i (element of selection vector) is either 1 (selected) or 0 (not selected) for CMAC_GBF, but in S_CMAC_GBF each element in the hypercube selection vector is equal to 1.

Fig. 5–The CMAC control scheme


169

The difference of S_CMAC and SOCM18 is that SOCM can create a new layer and delete a layer. The first process of the structure learning is to determine whether to add a new layer in association memory and to create its hypercube and weight memory, simultaneously. In the layer generating process, the mathematical description of the existing layers can be expressed as clusters. The firing strength of a rule for each incoming data can be represented as the degree to which the incoming data belong to the cluster. If a new input data falls within the boundary of clusters, SOCM will not generate a new layer but update parameters of the existing rules. Before SOCM decides which layer should be added or deleted, we have to choose the upper threshold (UT) (layer should be added) and lower threshold (LT) (layer should be deleted). The error rate can be expressed as

Erate = )1(

)(

tE

tE … (17)

If Erate>UT then nk(t+1) = nk(t)+1 … (18)

where nk(t) is the number of the existing layers at time t. In the meanwhile, for the new layer, the initial mean and variance of Gaussian basis function in association memory space are defined as mink = xs and σink(t) =σink(t-1), respectively.

If Erate<LT then nk(t+1) = nk(t)-1 … (19)

We choose the least contribution layer of the SOCM output to be deleted. Using the ratio of the output from every layer

LOi = SOCM

sii

v

xbv )(. , i = 1, 2,…nk … (20)

The layer which has minimum output (mini=1~nk

LOi) should be deleted.

Adaptive Learning Rule In order to improve the shortcoming of

conventional CMAC_GBF on the crisp relation, adaptive learning rule is introduced. Here, we use the discrete-time Lyapunov function to define the adaptive learning rate . Let the tracking error es be

es(t) = ss yy ˆ … (21)

where t is the time index. A discrete-type Lyapunov function can be expressed as

V = 2

1es

2(t) … (22)

Thus, the change in the Lyapunov function is obtained by

)()1(2

1)()1( 22 tetetVtVV ss … (23)

The error difference21 can be represented by

( ) ( 1) ( )

( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )

( )

s s s

i

s s sij

i ij ij

ij

e t e t e t

v te t e t e t

m tv t m t t

t

… (24)

From (14) to (16) we have

)()(

)(, siis

i

i

i

s

i

s xbav

w

w

y

y

e

tv

te

… (25)

,, 2

( )

( )

2( )( )

ijs s i

ij i ij ij

s i ijs i i i s

ij

e t e wy

m t y w m

x ma v b x

… (26)

2,

, 3

( )

( )

2( )( )

ijs s i

ij i ij ij

s i ijs i i i s

ij

e t e wy

t y w

x ma v b x

… (27)

From (23) to (25), with respect to vi, we have

)()(2)(2

1

)()1()()1(2

1

)()1(2

1 22

tetete

tetetete

teteV

sss

ssss

ss

)()()(

)(2

1)()(

,, siisse

vsiis

sss

xbateN

xba

tetete

, ,

2

,

2

,

1( ) ( ) ( ) ( )

2

1( ) ( )

2

2 ( ) ( ) ( )

vs s i i s s s i i s

e

vs s i i s

e

vs s s i i s

e

e t a b x e t a b xN

e t a b xN

e t e t a b xN


170

22,

2

,

1( ) ( )

2

2 ( )

vs s i i s

e

vs i i s

e

e t a b xN

a b xN

… (28)

Let 0)(2 2,

siis

e

v xbaN

then 0V , i.e., we can select the learning rate αv in the following range

0)(

22

,

v

siis

e

xba

N … (29)

From (23), (24), and (26), with respect to mij, we have

2 2

2

,

, 2

2

,, 2

2,

1( 1) ( )

21

( ) ( ) ( )2

2( )1( ) ( )

2

2( )2 ( ) ( ) ( )

2(1( ) ( )

2

s s

s s s

s i ijms s i i i s

e ij

s i ijms s s i i i s

e ij

sms s i i i s

e

V e t e t

e t e t e t

x me t a v b x

N

x me t e t a v b x

N

xe t a v b x

N

2

,

2

2

,

, 2

)

2( )2 ( )

i ij

ij

s i ijms i i i s

e ij

m

x ma v b x

N

… (30)

Let 0)(2

)(2

2

2

,,

ij

ijissiiis

e

mmx

xbvaN

then 0V , i.e., we can select the learning rate αm in the following range

2

,

, 2

20

2( )( )

em

s i ij

s i i i s

ij

N

x ma v b x

… (31)

From (23), (25), and (27), with respect to ij , we

have

2 2

22,

, 3

22,

, 3

1 1( 1) ( ) ( ) ( ) ( )

2 2

2( )1( ) ( )( )

2

2( )2 ( ) ( ) ( )( )

s s s s s

s i ijs s i i i s

e ij

s i ij

s s s i i i se ij

V e t e t e t e t e t

x me t a v b x

N

x me t e t a v b x

N

2

3

2,

,

2

3

2,

,2

))(2

)((2

))(2

)(()(2

1

ij

ijissiiis

e

ij

ijissiiiss

e

mxxbva

N

mxxbvate

N

… (32)

Let 0))(2

)((2

2

3

2,

,

ij

ijissiiis

e

mxxbva

N

then 0V , i.e., we can select the learning rate ασ in the following range

0

))(2

)((

22

3

2,

,

ij

ijissiiis

e

mxxbva

N … (33)

V becomes negative definite. This implies that 0)( tE for t . The convergence of the adaptive

SOCM learning process is then guaranteed. The aircraft landing control system is locally asymptotically stable.

Optimization Algorithms

The proposed control scheme still needs a PID controller to provide primary control signals to the pitch autopilot. Instead of trial-and-error, PID control gains are tuned by different artificial life models as shown in Fig. 6. In this study, BFO, PSO, CPSO, and BSO are applied to tune the control gains.

Fig. 6–Tuning PID control gains


171

BFO The flowchart of the BFO is shown in Fig. 7. The

bacterial foraging system consists of four principal mechanisms, namely chemotaxis, swarming, repro-duction, and elimination dispersal4.

Chemotaxis

This process simulates the movement of an Escherichia coli (E. coli) cell through swimming and tumbling via flagella. Biologically, an E. coli bacterium can move in two different ways. It can swim for a period of time in the same direction, or it may tumble, and alternate between these two modes of operation for the entire lifetime. Suppose θi (j,k,l) represents ith bacterium at jth chemotactic, kth reproductive and lth elimination dispersal step. C(i) is the size of the step taken in the random direction specified by the tumble (run length unit). Then in computational chemotaxis the movement of the bacterium may be represented by

)()(

)()(),,(),,1(

ii

iiClkjlkj

T

ii

… (34)

where indicates a vector in the random direction whose elements lie in [–1, 1]. Swarming

An interesting group behavior has been observed for several motile species of bacteria including E. coli and Salmonella Typhimurium (S. typhimurium), where intricate and stable spatiotemporal patterns (swarms)

are formed in a semisolid nutrient medium. A group of E. coli cells arrange themselves in a traveling ring by moving up the nutrient gradient when placed amidst a semisolid matrix with a single nutrient chemoeffecter. The cells, when stimulated by a high level of succinate, release an attractant aspertate, which helps them to aggregate into groups and thus move as concentric patterns of swarms with high bacterial dtion Aensity. The cell-to-cell signaling in E. coli swarm may be represented by the following equation:

)),,(,()),,(,(1

lkjJlkjPJ is

icccc

s

i

p

m

immtattractattracd

1 1

2tantan )])(exp([

2

1 1

[ exp( ( ) )]ps

irepellant repellant m m

i m

h

… (35)

where Jcc(θ, P(j,k,l)) is the objective function value to be added to the actual objective function (to be minimized) to present a time-varying objective function, S is the total number of bacteria, p is the number of variables to be optimized that are present

in each bacterium, and θ = [θ1,θ2,...,θpT is a point in

the p-dimensional search domain. tattracd tan , tattrac tan ,

repellanth , repellant are different coefficients that should be

chosen properly. Reproduction

The least healthy bacteria eventually die while each of the healthier bacteria (those yielding lower value of the objective function) asexually split into two bacteria, which are then placed in the same location. This keeps the swarm size constant.

Elimination and Dispersal

Gradual or sudden changes in the local environ-ment where a bacterium population lives may occur due to various reasons, e.g., a significant local rise of temperature may kill a group of bacteria that are currently in a region with a high concentration of nutrient gradients. Events can take place in such a fashion that all the bacteria in a region are killed or a group is dispersed into a new location. To simulate this phenomenon in BFO, some bacteria are liquid-dated at random with a very small probability while the new replacements are randomly initialized over the search space. The flowchart is given in Fig. 7.

Fig. 7–Flowchart of the bacterial foraging algorithm


172

PSO The flowchart of the PSO is shown in Fig. 8. The

main steps in the particle swarm optimization process are described as5: (i) Initialize a population (array) of particles with

random positions and velocities in the problem space.

(ii) Calculate the fitness function and set the values to the pbest for each particle, and set the best value of all the particles to gbest.

(iii) Change the velocity and position of the particle according to Eqs (36) and (37), respectively:

)(1 )(1

)()1( kidid

kid

kid xpbestrandcvwv

)(2 )(2

kidd xgbestrandc … (36)

)1()1( kid

kid

kid vxx … (37)

(iv) Calculate the fitness again and compare the particle’s fitness evaluation with the particle’s pbest. If the current value is better than pbest, then set the pbest value equal to the current value, and the pbest location equal to the current location in d-dimensional space.

(v) Compare the fitness evaluation with the population’s overall previous best. If the current value is better than gbest, then reset gbest to the current particle’s array index and value.

(vi) Loop to step (iii) until a criterion is met, usually a sufficiently good fitness or a maximum number of iterations (generations).

The definition of the parameters are:

)( kidv : velocity of individual i at iteration k,

max)(mind

kidd VvV

w : inertia weight factor, ,1c 2c : acceleration constant,

rand1, rand2 : uniform random number between 0 and 1,

)( kidx : current position of individual i at

iteration k ,

ipbest : pbest of individual i, gbest : gbest of the group.

CPSO

The random movement which is obtained by the definite motion equation is called chaotic motion. The variable that presents chaotic state is called chaotic variable that is very sensitive to initial value. The chaotic variable has the characteristic of traversing all states according to the determination formula from any point (except for fixed-point) starting. The Logistic equation is a typical chaotic map system, its expression6 is:

)1(1 nnn XXX , n=1,2,3,...,

]1,0[,40 nX … (38)

where, μ is a control parameter. According to chaotic variables with ergodicity and randomness having the enhancement population's search ability, based on chaotic ideas, some scholars have proposed many kinds of improvement chaos particle swarm optimization (CPSO) algorithms. The improvement algorithm's fundamental mode is:

(i) using chaotic variables to initialize particle's position and velocity in order to increase population's diversity and ergodicity;

(ii) using early-maturing judgment mechanisms to monitor population's evolution situation, and randomly generating an initial value to replace original swarm in order to reinitialize the particle's velocity.

Since these chaotic variables in CPSO algorithm has not involved in the parameters such as inertia weight w and random numbers 1c and 2c of PSO algorithm, so these parameter choices will have a very tremendous influence to optimize performance of CPSO algorithm. Especially, when the w value is larger, which will help PSO to jump out the local optimum, and when the w value is smaller, which will be advantageous to CPSO algorithm convergence.

Fig. 8–Flowchart of the particle swarm optimization


173

BSO In what follows we briefly outline the new BSO

algorithm step by step7.

(i) Initialize parameters n, N, NC, NS, Nre, Ned,

Ped, C(i)( i=1,2,…,N), i ,

where

n: Dimension of the search space, N: The number of bacteria in the population, NC : No. of Chemo-tactic steps, Nre: The number of reproduction steps, Ned: The number of elimination- dispersal events, Ped: Elimination-dispersal with probability, C(i): The size of the step taken in the random

direction specified by the tumble, ω: The inertia weight, C1: Swarm Confidence,

),,( kji : Position vector of the ith bacterium, in jth

chemotactic step, and kth reproduction,

iV : Velocity vector of the ith bacterium.

(ii) Update the following parameters: J(i, j, k): Cost or fitness value of the ith bacterium in

the jth chemo-taxis, and kth reproduction loop.

bestg _ : Position vector of the best position found by

all bacteria. Jbest(i, j, k): Fitness of the best position found so far. (iii) Reproduction loop: k=k+1 (iv) Chemotaxis loop: j=j+1 [substep_a] For i =1, 2,…,N, take achemotactic step for bacterium i as follows. [substep_b] Compute fitness function, J(i ,j, k). [substep_c] Let Jlast=J(i,j,k) to save this value since we may find a better cost via a run. [substep_d]Tumble: generate a random vector

nRi )( with each element )(im , m=1,2,..., p, a

random number on [-1, 1]. [substep_e] Move: let

)()(

)()(),,(),1,(

ii

iiCkjikji

T

… (39)

[substep_f] Compute J (i, j +1, k) . [substep_g] Swim: we consider only the ith bacterium is swimming while the others are not moving then

i) Let m=0 (counter for swim length).

ii) While m< s N (if have not climbed down too long). • Let m=m+1. • If J(i, j +1, k) < Jlast (if doing better), let Jlast = J

(i,j +1, k) and let

)()(

)()(),,(),1,(

ii

iiCkjikji

T

… (40)

and use this θ(i, j +1, k) to compute the new J (i, j +1, k) as we did in [substep_f] • Else, let m= sN . This is the end of the while

statement. (v) Mutation with PSO operator For i = 1,2, …, S Update the

bestg _ and bestJ (i, j, k)

Update position and velocity of the dth coordinate of the ith bacterium according to the following rule:

1 1

_( ( , 1, ))d

new newid id

oldg best d

V V C

i j k

new

idoldd

newd Vkjikji ),1,(),1,( … (41)

(vi) Let S_r = S/2. The S_r bacteria with highest cost function (J)

values die and the other half of bacteria population with the best values split (and the copies that are made are placed at the same location as their parent).

(vii) If k<Nre, go to step (i). We have not reached the specified number of reproduction steps. So we start the next generation in the chemo-taxis loop.

Simulations

The aircraft starts the initial states of the ALS as: the flight height is 500 ft, the horizontal position before touching the ground is 9240 ft, the flight angle is -3°, the speed of the aircraft is 234.7 ft/s. Successful touchdown landing conditions are defined as:

-3 TDh ft/s 0,200 )(TVTD ft/s 270,

-300 TDx ft 1000, -10 TD degree 5,

where T is the time at touch down, TDh is the vertical

speed of the aircraft at touch down, TDx is the position

at touch down, )(tVTD is horizontal speed, TD is the pitch angle at touch down.

For the safe landing of an aircraft using different CMACs control schemes under different turbulence speeds are tested. The simulations show that the


174

adaptive SOCM controller has good adaptive capability against severe wind disturbance and is more robust than CMAC, CMAC_GBF, S_CMAC_GBF, and PID controllers, as shown in Table 1. The adaptive SOCM control scheme can

successfully guide the aircraft flying through wind speeds of 0 ft/s to 96 ft/s as shown in Figs 9-12. Different combinational control schemes are also tested. Figures 13-16 show the results of using BSO

Fig. 11–Aircraft vertical velocity and command

Fig. 12–Aircraft altitude and command

Fig. 13–Turbulence profile (148 ft/sec)

Table 1– Results from using different controllers

Controller Wind speed (ft/s)

Landing point (ft)

Vertical speed (ft/s)

Pitch angle

(°) PID 30 937.80 -2.19 -0.48 CMAC 58 843.93 -2.57 -9.82 CMAC_GBF 82 984.73 -1.08 2.01 S_CMAC_GBF 78 656.18 -1.53 1.84 SOCM 96 961.26 -1.400 1.51

Fig. 9–Turbulence profile (96 ft/sec)

Fig. 10–Aircraft pitch and pitch command


175

Fig. 14–Aircraft pitch and pitch command

Fig. 15–Aircraft vertical velocity and command

Fig. 16–Aircraft altitude and command

Table 2 – Results from using CMAC with different optimization algorithms

Optimization Wind Landing Vertical Pitch Algorithms speed point speed angle (ft/s) (ft) (ft/s) (°) BFO 132 444.978 -2.0992 2.6035 PSO 100 574.0509 -2.019 3.5907 CPSO 136 304.1713 -1.9881 3.5485 BSO 148 421.5102 -2.1695 4.532

algorithm with CMAC controller in the wind turbulence speed at 148 ft/s. Table 2 shows the maximum wind speed that the proposed automatic landing controller can overcome by using different optimization algorithms with the CMAC in turbulence condition. Conclusions

This paper investigates the uses of PID controller with optimization algorithms (BFO, PSO, CPSO, and BSO), CMAC, CMAC_GBF, SCMAC_GBF and adaptive SOCM in the aircraft automatic landing control and to make the automatic landing system more intelligent. Tracking performance and adaptive capability are demonstrated through software simulations. The conventional PID controller can only overcome about 30 ft/s turbulence speed. With optimal gains, the proposed controller has better adaptive capability than conventional PID controller. The performance of CMAC_GBF is better than CMAC controller. Furthermore, the addressing technique is simplified to speed up the computation of the intelligent controller. The result of adaptive SOCM simulation is great and the computing time is less which is suitable for on-line control system.

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