1

Fuzzy Sliding Mode Autopilot Design for Nonminimum Phase and Nonlinear UAV

A.R. Babaei1 ,M. Mortazavi2

Aerospace Department, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

M.H. Moradi3

Medical Engineering Department, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

Abstract

The fuzzy sliding mode control based on the multi-objective genetic algorithm is proposed to design an altitude autopilot of UAV. This case presents an interesting challenge due to the non-minimum phase characteristic, nonlinearities and uncertainties of the altitude to elevator relation. The responses of this autopilot are investigated through various criteria such as, the time response characteristics, robustness with respect to parametric uncertainties, and robustness with respect to unmodeled dynamics. The parametric robustness is investigated with reduction in significant longitudinal stability coefficients. Also, a nonlinear model in presence of the coupling terms is used to investigate the robustness with respect to unmodeled dynamics. In spite of a designed classic autopilot, it is shown by simulation that combining of the sliding mode control robustness and the fuzzy logic control independence of system model can guarantee the acceptable robust performance and stability with respect to unmodeled dynamics and parametric uncertainty, while the number of FSMC rules is smaller than that for the conventional fuzzy logic control.

Keywords: Autopilot, UAV, Sliding Mode Control, Fuzzy Logic Control, Uncertainty, Nonminimum Phase

Nomenclature FSMC = Fuzzy Sliding Mode Control FLC = Fuzzy Logic Control SMC = Sliding Mode Control I/O = input/output GA = genetic algorithm UAV= unmanned aerial vehicle Eδ = elevator control variable

trimEδ = elevator trim angle Rδ = rudder control variable Aδ = aileron control variable h = UAV altitude ch = Command altitude

1 PhD student, Fax: +982166404885/EMail: arbabaei@aut.ac.ir, corresponding author. 2Assistant professor, Fax: +982166404885/EMail: mortazavi@aut.ac.ir. 3Associated professor, Fax: +98216640 /EMail: mhmoradi@aut.ac.ir.

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φ = roll angle θ = pitch angle ψ = heading angle P = roll angle rate Q = pitch angle rate R = heading angle rate U = forward velocity W = vertical velocity V = lateral velocity

tV = total velocity α = angle of attack β = side slip angle

popN = the number of chromosome population

varN = the number of a chromosome variables

keepN = the number of kept chromosome for mating

0β = a random number on the interval [0, 1] mnp = nth variable in the mother chromosome dnp = nth variable in the father chromosome σ = standard deviation of the normal distribution (0,1)nN = the standard normal distribution (mean = 0 and variance = 1)

ξ = damping ratio

nω = natural frequency λ = a strictly constant number N = number of rules ( )A xµ = membership function of A set at x value

1. Introduction

The main difficulties in aerospace vehicles autopilot design are known to result from the

aerodynamic parameter uncertainties. Because the knowledge of aerodynamic coefficients as

well as their dependencies on some parameters (e.g. angle of attack, side slip angle… etc.) is

very imprecise, the design controller must not be sensitive to the variations of these

coefficients. In general, the dynamics of aerospace vehicles are nonlinear, time varying,

uncertain and they are controlled by assuming that inertial cross couplings among roll, pitch,

and yaw dynamics are negligible. In practice, there is a coupling among the dynamics, and

this can result in performance degradation. To design a high performance autopilot, it is

desirable to use a more general model containing the coupling terms. Therefore, one of major

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problems in designing a flight control system is modeling uncertainties such as, parameters

variation in characterizing an aerospace vehicle model and unknown nonlinear dynamics.

In order to get ride of the exact model restrictions, several adaptive schemes have been

introduced to solve the problem of linearly parameterized uncertainties [2, 11], which are

referred to as structured uncertainties. Unfortunately, in the some of applications, there are

some controlled systems which are characterized by an unmodeled or/and unknown dynamics

which are referred the unstructured uncertainties.

Feedback linearization [20] is a popular method used in nonlinear control applications, and

there have been several flight control demonstrations [1, 5, 16, and 21]. This technique has

received much of attention and shows great promise. The main drawback to the nonlinear

control approach such as feedback linearization is that, as model-based control method, they

require accurate knowledge of the plant dynamics. This is significant in flight control since

the aerodynamic of parameters always contain some degree of uncertainty.

In this paper, a proposed technique is applied to the UAV altitude control which is

aerodynamically controlled by tail control surfaces using single-loop scheme. The

uncertainties that are considered are due to both parametric uncertainty and unknown

nonlinear effects specially, high coupling among pitch and yaw channels. In fact, the I/O

feedback linearization technique which has been regarded as the powerful design method for

nonlinear systems, can not directly applied to altitude control of UAV because the pitch

channel of tail-controlled UAV is inherently nonminimum phase. These fact make it difficult

to design high performance autopilot.

In this paper, it is tried to achieve many properties: a desirable tracking, a suitable time

response characteristics, robustness with respect to unmodeled dynamics such as coupling

effects, and a robustness with respect to parametric uncertainty. Two strategies are utilized to

design an autopilot. In first strategy, the classic compensator that utilizes the root locus

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method capabilities is designed based on linear model. In second strategy, the sliding mode

control method [20] that the system uncertainties and external disturbances can be handled,

the fuzzy logic [23] capabilities that are independent of system model and the GA optimum

search mechanism [8] have been used in designing the autopilot that called FSMC. The

FSMC method has recently been used for various applications [6, 13, 15, and 18] and it can

be used as the robust method for aerospace applications. The FLC method has been used for

various cases [10, 12, and 25]. Also, optimal determination of fuzzy systems properties has

been widely used in different applications by evolutionary algorithms [4, 14, 19, and 24]. In

[12], the UAV altitude based on linear model is controlled by the expert knowledge-based

FLC so that, robustness with respect to parametric uncertainty is achieved. In ref. [4] and [22]

the missile acceleration that is nonminimum phase is indirectly controlled (lateral velocities

are controlled) by FLC and GA. In ref. [2], the UAV altitude with linear model is controlled

by combination of a linear controller and a simple adaptive controller to modify robustness

with respect to parametric uncertainties only. In ref. [14], the fuzzy PID controller based on

GA for nonminimum phase and linear systems is presented to eliminate undershoot that it is

due to unstable zeros.

The rest of the paper is organized as follows: Section 2 presents the linear and nonlinear

model of the UAV. The linear autopilot design is presented in section 3. The FSMC based on

GA autopilot is designed in Section 4. The Simulation results and their analytic discussions

are presented in section 5 to investigate several aspects. Finally, conclusions are drawn in

Section 6.

2. UAV Model

The nonlinear motion equations of UAV are derived as follows:

EU=-9.8sin -QW+RV-0.0125U-16.63 +16.6 4.5θ α δ +ɺ (1-a)

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AV=9.8sin cos +PW-RU-263.7 -0.0053P+1.64R-0.0032 58.2 Rφ θ β δ δ−ɺ (1-b)

EW=9.8cos cos +QU-PV-0.068U-259 -1.3Q+57.5 21φ θ α δ +ɺ (1-c)

A RP=-1.51QR+0.04PQ+76.7 -1.9P-0.68R+149 +105β δ δɺ (1-d)

2 2Q=1.03PR-0.017(P ) 988 8.9 1362 0.284ER Qα δ− − − + −ɺ (1-e)

AR=-0.038QR-0.85PQ+306 -0.044P-2.82R+2.27 434 Rβ δ δ+ɺ (1-f)

cos sin

cost

W U

V

α αα

β

−=ɺ ɺ

ɺ (1-g)

1cos sin cos sin sin

t

U V WV

β α β β α β = − + − ɺ ɺ ɺ ɺ (1-h)

sin tan cos tanP Q Rφ φ θ φ θ= + +ɺ (1-i)

cos sinQ Rθ φ φ= −ɺ (1-j)

( )sin cos secQ Rψ φ φ θ= +ɺ (1-l)

sin cos sin cos cosh U V Wθ θ φ θ φ= − −ɺ (1-m)

2 2 2tV U V W= + + (1-n)

δ δ δ= +trimE E e (1-o)

Some of the UAV variables are defined in Fig. (1). We are aimed to design the altitude

hold mode autopilot for this UAV. An altitude hold mode is an autopilot mode which tries to

maintain altitude by the elevator input. The nominal linear model for altitude is written as

following transfer function:

(2) ( )( )( )

( )2 2

( ) -57.3 s-24.6 21 .008

( ) s(s +0.011s+0.0022) s 2.12 98.4δ

+ +=

+ +e

h S s s

S s

It can be received that the altitude output and elevator input relation is non-minimum

phase, and the longitudinal flight modes aren’t desirable. If 40% reduction is applied to

, ,q um m DC C C

α, and

uLC (see Table I [3]), a degraded linear model is presented as following

transfer function to investigate the parametric robustness.

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(3) ( )( )( )

( )2 2

( ) -57.3 s-25.5 21.6 .0017

( ) s(s +0.0055s+0.0021) s 1.82 64δ

+ +=

+ +e

h S s s

S s

These variations in plant dynamics are caused moving an unstable zero further to the right,

and also caused degradation of the phoguid and the short period flight modes.

3. Linear Autopilot Design

In this section, a classic autopilot is designed for ψ and h variables using the nominal

linear model. However, the ψ -autopilot isn’t presented and assumed that there aren’t

uncertainties in directional-lateral channel. The block diagram of the altitude hold mode to

design a compensator is shown in Fig. (2).

Two compensators are designed by the root locus techniques as following transfer

functions:

(4-a) 2

2

0.00681(S+0.1) (S +2.12S+98.4)

(S+20) (S +6.94S+13.1)=CaG

(4-b) 2

2

0.012(S+0.05) (S +2.12S+98.4)

(S+20) (S +6S+15.25)=

bCG

For closed loop system, the dominant characteristics are attained by applying these autopilots

to the nominal linear model (2) as following values.

n n: 0.79, 1.36, : 0.6, 2.31a bξ ω ξ ω= = = =

To evaluate the designed autopilots, those are applied to the nominal linear model, the

degraded linear model and the nonlinear model. The commands are cψ =0, 10 degree and

hc=10 m. The simulation results are illustrated in Fig. (3) and Fig. (4). According to Fig. 3, the

desirable response is seen for the nominal linear model. Concerning to degraded linear model,

relative parametric robustness of first controller and low parametric robustness of second

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controller is known. By inspection of Fig. 4, the time response of both autopilots has been

degraded for the UAV nonlinear model because the nonlinear terms have been appeared.

Clearly, increasing the ψ–command value lead to increase the degraded coupling nonlinear

effects. Therefore, the compensators can’t meet the wanted requirements in term of the

parametric uncertainties and the unmodeled dynamics. Now, eliminating the nonlinear terms

effects is required to achieve the desirable tracking. In addition, the autopilot should not be

very sensitive to the variation of system parameters. Due to these reasons, it is tried to exploit

the knowledge-based FLC that is independent of system model.

It should be noted that the elevator trim angle (here, 1.92trimEδ = � ) must be added to the

control input in nonlinear simulation procedure because the controllers have been designed

based on the nominal linear model.

4. FSMC Autopilot Design

The SMC [20] is a robust design methodology using a systematic scheme based on a

sliding mode surface and Lyapunov stability theorem. The main advantage of the SMC is that

the system uncertainties and external disturbances can be handled under the invariance

characteristics of system’s sliding mode state with guaranteed system stability.

Fuzzy systems are knowledge-based or rule-based systems that were initiated by Lotfi A.

Zadeh in 1965 [26]. The heart of a fuzzy system is a knowledge consisting of the so-called

fuzzy IF-THEN rules. A fuzzy IF-THEN rule is an IF-THEN statement in which some words

are characterized by continuous membership functions. A block diagram of a fuzzy logic

system is shown in Fig. (5).

The SMC theory uses discontinuous control action to drive state trajectories toward a

specific surface until stable equilibrium state is reached. This principle provides guidance to

design a fuzzy logic controller.

Consider the following single-input nth order nonlinear system:

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(5)

1 2

2 3

( ) ( )

( ) ( )

( ) ( ) ( )n

x t x t

x t x t

x t f X b X u

=

=

= +

ɺ

ɺ

⋮

ɺ

Where 1 2[ ]TnX x x x= … is the state vector, and u is the control input. If the desired state

vector is defined as 1 2[ ]Tc c c cnX x x x= … , then the error vector 1 2[ ]TnE e e e= … could be

written as

(6) 1 1 2 2

T

c c c n cnE X X x x x x x x = − = − − − …

And a linear function : ns →ℝ ℝ is defined as

(7) s CE=

Where 1 2 nC c c c = … is the coefficient row vector, andic 's are all real numbers. Then, a

sliding surface can be represented as

(8) 1 1 2 2 0n nS c e c e c e= + + + =⋯

To design the control input ( )u t so that the state trajectories are driven and attracted toward

the sliding surface and then remain sliding on it, the following inequality must be satisfied

(9) ss sη< −ɺ

The idea behind equation (9) is in the sense of Lyapunov function. If we treat s as a scalar

time function and the Lyapunov function ν as

(10) 210

2sν = >

Together with (9), we can guarantee that asymptotic stability of the system is satisfied.

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Here, the altitude control is concerned. For this reason, we can use the equations (1-a)-(1-

o) as follow

(11)

( )

1 2

2 1 1 1 2 2

( ) ( )

( ) ( ) ,

X F X B X

x x

x f X b X X

δ

δ δ

= +

=

= + +∆

ɺ

ɺ

ɺ

Where, 1 2[ ]T T TX X X= , 1 [ ]TX U W Q θ α= 2 [ ]TX V P R φ β= , 1 2[ ]T T Tδ δ δ= , 1 Eδ δ= ,

2 [ ]TRAδ δ δ= , 1x h= , and ( )2 2,X δ∆ is contribution of lateral-directional variables that we

call it coupling term. Specially, the ∆ term is nearly zero when the cψ and cφ are zero. The

( )F X , 1( )f X , ( )B X , and 1( )b X are the continuous linear or the nonlinear functions that they are

derived by equations (1-a)-(1-o).

By concerning to altitude control significance in this paper, the sliding function is defined

as follow:

(12) 1 1 1 1( ) ( )c cs x x x xλ= − + −ɺ ɺ

And the sliding surface is presented as follow:

(13) 1 1 1 1( ) ( ) 0c cS x x x xλ= − + − =ɺ ɺ

By concerning to equations (9) and (11) we have following relation.

(14)

[ ]

( )

1 1 1 1

1 2 2 1 1

0 1 1

( ) ( )

( ( ) , ) ( )

( )

c c

E c

E c

ss s x x x x

s f X X b X s sx se

s F b X s sx se

λ

δ δ λ

δ λ

= − + −

= +∆ + − +

≤ + − +

ɺ ɺɺ ɺɺ ɺ ɺ

ɺɺ ɺ

ɺɺ ɺ

Where 1 1ce x x= − and ( )1 2 2 0( ) ,f X X Fδ+∆ ≤ .

Therefore, we can drive following results:

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- The control input on the two sides of the sliding surface are opposite in sign and its

magnitude is proportional to the sliding function (12).

- If the term b is negative, then ( ) ( )Esign sign sδ = .

We can conclude thatE sδ ∝ for negativeb . This can be used to derive a fuzzy sliding mode

controller. Therefore, the following fuzzy sliding mode control is designed to obtain the

control input:

(15) : l l E lR IF s IS S THEN IS Uδ

Where lS is the linguistic value of s in the l th -fuzzy rule, and lU is the linguistic value of Eδ

in the l th -fuzzy rule.

Here, the b term is derived as follow:

(16) 16.6 sin 57.5 cosb θ θ= −

For 74θ < � , the 2b term is negative. For our case, the condition 74θ < � is always satisfied.

Therefore, we can tell that the b term is always negative. Based on these discussions, the

rules base, for FSMC design, is presented in table (2), so that the gaussian membership

functions for the sliding mode function are used (see Fig. (6)).

Suppose that the fuzzy set lU and lS in (15) are normal with centerlu and ls . Then the

fuzzy systems with rule base (15), product inference engine, singleton fuzzifier, and center

average defuzzifier are of the following form [23]:

(17)

21

1

( )

, ( ) exp2

( )

l

l

l

Ml

Sll

E SMl

S

l

u ss s

s

s

µ

δ µσ

µ

=

=

− = = −

∑

∑

Here, 7M = .

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Traditionally, the FLC is designed by expert knowledge and experience. It is difficult to

decide the control rules as system is complex, for instance, a nonminimum phase.

Increasingly, the GA is used to facilitate the FLS design so that, it can be used to design the

FSMC. The GA is a general-purpose search algorithm that uses principles inspired by natural

population genetics to evolve solutions to problems. It was first proposed in ref. [9]. The GA

is theoretically and empirically proven to provide a robust search in complex spaces, thereby

offering a valid approach to problems requiring efficient and effective searches [7].

Here, the continuous or real-valued GA is used [8]. A path through the components of the

GA is shown as a flowchart in Fig. (7). First, a chromosome population (popN ) is randomly

generated. Each chromosome specifies a candidate solution of the optimization problem. The

fitness of all individuals with respect to the optimization task is then evaluated by a scalar cost

function (fitness function).

The cost function is a surface with peaks and valleys when displayed in variable space,

much like a topographic map. To find a valley, an optimization algorithm searches for the

minimum cost. To find a peak, an optimization algorithm searches for the maximum cost. A

cost function generates an output from a set of input variables (a chromosome). The object is

to modify the output in some desirable fashion by finding the appropriate values for the input

variables.

If the chromosome has varN variables given by var1 2, , , Np p p… then the chromosome is

written as an varN element row vector. Then is the time to decide which chromosomes in the

initial population are fit enough to survive and possibly reproduce offspring in the next

generation. The popN costs and associated chromosomes are ranked from lowest cost to

highest cost. From the popN chromosomes in a given generation, only the top keepN are kept

for mating (process of natural selection) and the rest are discarded to make room for the new

offspring.

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One mother and one father in some random fashion are selected. Each pair produces two

offspring that contain traits from each parent. In addition the parents survive to be part of the

next generation.

One crossover point is randomly selected, and then the blending methods are applied by

finding ways to combine variable values from the two parents into new variable values in the

offspring. A single offspring variable value,newp , comes from a combination of the two

corresponding offspring variable values [17].

(18) 0 0(1 )new mn dnp p pβ β= + −

If care is not taken, the GA can converge too quickly into one region of the cost surface. If

this area is in the region of the global minimum, that is good. However, some functions have

many local minima. If the tendency to converge quickly is not solved, a local rather than a

global minimum is attained. To avoid this problem of overly fast convergence, it is forced the

routine to explore other areas of the cost surface by randomly introducing changes, or

mutations, in some of the variables. Most users of the continuous GA add a normally

distributed random number to the variable selected for mutation:

(19) (0,1)n n np p Nσ′ = +

The centers of output membership functions (lu ), λ , and variance (iσ ) for input

membership functions are considered as a chromosome. The input variable boundaries are

12 , 12up lowu u= = −� � , and for the sliding mode function, these are 100, s 100up lows = = − .

Here, to determine the FSMC parameters, the GA with following properties is used:

• Chromosome population, popN = 50

• Number of generation = 50

• Mutation rate = 2%

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• keepN = 50%

• Cost function, 1 1 2 2 3 3J W S W S W S= + + , so that iW is corresponding weight (see Fig. (8)):

1

10

t

cS h h dt= −∫ , this parameter covers rise time and undershoot that is caused by

nonminimum phase effect.

2

12

t

ct

S h h dt= −∫ , this parameter includes the overshoot value.

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T

ct

S h h dt= −∫ , this parameter comprises the settling time, the steady stat error, and

unstable effects.

The parameters in Fig. (8) are defined as follow:

1t is time of first intersection between altitude time response and altitude command, 2t is time

of second intersection between altitude time response and altitude command, and T is final

time of simulation. If the intersections don't create, the it parameter is considered zero value.

In this alternative form of the cost function, the undershoot, the overshoot, the rise time,

the settling time, the steady stat error, and the stability objectives are easily covered to

determine the optimal FSMC. Therefore, computation of all objectives isn’t individually

required.

By consideration a trade-off procedure between time response characteristics and

robustness, the weights as 1 2 31, 1.4, 1.3W W W= = = are chosen.

The closed loop scheme for this strategy is illustrated in Fig. (9). Due to the computational

requirements, the controller is generally evolved off-line (using GA and a model of the

controlled process). For this reason, the nominal linear model, as an available mathematical

model of UAV, is utilized to achieve the controller parameters. Then, using the nonlinear

model in simulation procedure, the robustness of controller with respect to the unmodeled

dynamics that are not considered in the GA is investigated. Here, the nonlinear model is

assumed as a real model of UAV. The GA results based on the nominal linear model are

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presented in table (3). A simulation result for the nominal linear model is illustrated in Fig.

(10). This figure displays the suitable time response characteristics. Also, a mean fitness value

in each generation is shown in Fig.(11).

5. Results

In this section, the designed autopilots will be investigated in several aspects. First, the

robustness of FSMC is investigated. The simulation results are illustrated in Fig. (12) for the

degraded linear model and the nonlinear model. The desirable performance of FSMC in

presence of the parametric uncertainties and the unmodeled dynamics is clear. Therefore, the

FSMC with low rules and parameters is a robust method to design autopilots such as altitude

hold mode. Therefore, in spite of the comparable time response characteristics of classic

controllers and FSMC for the nominal linear model, the FSMC strategy is robust with respect

to parametric uncertainty and unmodeled dynamics. In Fig. (13), input behavior is illustrated

for three autopilots. The FSMC input is smaller than that for classic autopilots. Also, the

FSMC input has a desirable behavior.

Second, the performance of classic control and FSMC is investigated by reduction and

increasing of controllers gain. For instance, if we multiply signal of controllers by 0.5 and 2,

then the results is shown in Fig. (14). The time response of system in presence of the classic

autopilot is largely degraded. For multiplier 2, the overshoot is nearly 50% and for multiplier

0.5 the rise time is nearly 5 second. But, the performance degradation of FSMC is less than

that for classic autopilots, while these parameters are 30% and 2.7 second.

Thirdly, it is required that we investigate other flight variables such as velocity, angle of

attack, and pitch rate. The time response of these variables are illustrated in Fig. (15).

According to this figure, the velocity time response has desirable behavior for the FSMC, but

Angle of attack and pitch rate variables time response hasn't desirable behavior. The altitude

control lead to modify the phoguid mode (in this mode, velocity and altitude variables are

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significant), and we can not well modify the short period mode of aircraft (in this mode, angle

of attack and pitch rate variables are significant). This goal is feasible by using the inner

loops, or other states have been applied to sliding mode function and redefinition of cost

function in term of sliding mode function. In other work, we are going to use recent procedure

for other applications. In this application, the altitude hold mode is significant because the

unmanned aerial vehicles fly in near the terrain surface.

Comparison of three autopilots has been summarized in table (4). According to this table,

if model of UAV hasn't uncertainty, then classic autopilot is proposed to design the autopilots.

But, the FSMC is proposed to design the autopilots in presence of uncertainties.

6. Conclusions

In this paper, the altitude hold mode autopilot has been designed for an unmanned aerial

vehicle. This case presents the interesting challenge due to the non-minimum phase

characteristic, nonlinearities and uncertainties of the altitude to elevator relation. There are

nonlinear terms such as coupling among the dynamics that may lead to performance

degradation. The considered uncertainties are due to both parametric uncertainty and

unmodeled nonlinear terms. It has been tried to achieve some properties: the desirable time

response characteristics, robustness with respect to parametric uncertainty, and robustness

with respect to unmodeled dynamics. For this reasons, two strategies have been utilized to

design this autopilot. First, the compensator has been designed through the root-locus

techniques. Second, the fuzzy sliding mode controller has been designed through the multi-

objective genetic algorithm to mechanize its parameters determination. Both autopilots have

been designed based on the nominal linear model. The time response of both autopilots has

been investigated through above criteria i.e. the time response characteristics, robustness with

respect to parametric uncertainties, and robustness with respect to unmodeled dynamics. The

parametric robustness has been investigated with 40% reduction in significant longitudinal

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stability coefficients for linear model that called degraded linear model. To evaluate the

designed autopilots in term of robustness with respect to unmodeled dynamics, those are

applied to the nonlinear model that contains the coupling terms. The simulation results show

that the compensators can’t meet the wanted requirements in term of the parametric

uncertainties and the unmodeled dynamics. Increasing the ψ–command value lead to increase

the degraded coupling nonlinear effects. In the fuzzy sliding mode autopilot design procedure,

the alternative form of the cost function has been used to determine the optimal fuzzy

autopilot. In this cost function, the computation of all objectives isn’t individually required so

that the undershoot, the overshoot, the rise time, the settling time, the steady stat error, and the

stability objectives have been easily covered by it. The results show the fuzzy sliding mode

autopilot outperforms the compensator in terms of robustness with respect to parametric

uncertainties and robustness with respect to unmodeled dynamics while both have comparable

the time response characteristics for the nominal linear model. Also, after the control input

signal gain widely changed, the degradation amount of FSMC is less than that for the classic

autopilots. Finally, the fuzzy sliding mode control with low rules has high potential to design

the autopilot modes in presence of uncertainties. Also, based on a complex cost function, the

genetic algorithm gives us the optimal mechanization capability to design the fuzzy

autopilots.

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18

[12] C. Kelly and D.E. Bossert, Fuzzy Logic Non-minimum Phase Autopilot Design,

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19

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20

Table 1 List of Stability Derivatives that have the largest effects on the Dynamic Properties of longitudinal modes [3]

Stability Derivative

Quantity Most Affected

How Affected

qmC Damping of the Short Period, sξ

Increase qmC to

Increase the Damping mCα Natural Frequency

of the Short Period ,

snω

Increase mCα to

Increase the Frequency

uxC Damping of the

Phugoid, pξ Increase

uxC to

Increase the Damping uz

C Natural Frequency of the Phugoid,

pnω

Increase uz

C to

Increase the Frequency

21

Table 2 FSMC rules

Rule No. Rule

1 IF s NB THEN u IS NB

2 IF s NM THEN u IS NM

3 IF s NS THEN u IS NS

4 IF s ZE THEN u IS ZE

5 IF s PS THEN u IS PS

6 IF s PM THEN u IS PM

7 IF s PB THEN u IS PB

22

Table 3 The optimal parameters of the FSMC

NBsσ NMsσ

NSsσ ZEsσ

PSsσ PMsσ

PBsσ NMu NSu ZEu PSu PMu λ

23.6 28.4 16.8 19.8 17.5 26.9 20.8 -7.6 -4.73 0 4.34 9.5 0.84

23

Table 4 Comparison of the two strategies (three cases) performances

FSMC 2nd Comp.

1st Comp.

15 14 15 Overshoot (%) 2.6 1.6 2.3 Rise Time (Second)

desirable low almost

low Parametric Robustness

desirable low low Robustness with respect to Unmodeled Dynamics

suitable high high Degradation amount of time response in presence of changing input signal

24

Fig. 1 definition of UAV Parameters

Fig. 2 Single loop block diagram of the altitude hold mode autopilot to design a compensator

Fig. 3 Linear model time response with compensator

Fig. 4 Nonlinear model time response with compensator

Fig. 5 Rule-Based FLS

Fig. 6 Membership Functions for sliding mode function and Control Variable

Fig. 7 GA flowchart

Fig. 8 definition of S1 , S2 , and S3 parameters

Fig. 9 FSMC Scheme

Fig. 10 Nominal Linear model time response with FSMC-GA autopilot

Fig. 11 Mean fitness value for FSMC

Fig. 12 Degraded Linear model and nonlinear model time response with FSMC-GA

Fig. 13 Elevator time response

Fig. 14 Time response of system with changing a gain of autopilots

Fig. 15 Time response of velocity, angle of attack, and pitch rate variables

Fig. 16 state trajectory of FSMC

25

26

Ch Compensator Nominal Linear Model

20

s 20+

h + -

27

0 5 10 15 20 250

5

10

15

Alti

tude

,m

Nominal Linear Model

0 5 10 15 20 250

5

10

15

Time,s

Alti

tude

,m

Degraded Linear Model

1st Compensator

2nd Compensator

28

0 5 10 15 20 250

5

10

15

Alti

tude

,m

sai=0 deg

0 5 10 15 20 250

5

10

15

Time,s

Alti

tude

,m

sai=10 deg

1st Compensator

2nd Compensator

29

Crisp Input

Rules base

Fuzzifier Defuzzifier

Fuzzy Inference Engine

Crisp Output

Fuzzy Input Sets

Fuzzy Output Sets

30

s_low s_up0

1

s

MF s

fuzzy sliding mode control

u_low u_NM u_NS u_ZE u_PS u_PM u_up0

1

u

MF u

NB NMNS ZE PS PM PB

31

Generate Initial Population 1Gen =

Find cost for each chromosome (Evaluate Population)

Select Mates (Selection)

Crossover

Mutation

1+= GenGen

- Define cost function, Variables - Select GA Parameters

MaxGenGen >

Output Results

Yes

No

32

33

Dynamics

FSMC Actuator u

s e eλ= + ɺ ( )

ce h h

de e

dt

= −

=ɺ

ch h

34

0 5 10 15 20 250

2

4

6

8

10

12

time,second

altit

ude,

m

Nominal Linear Model and FSMC

35

0 10 20 30 40 5020

40

60

80

100

120

140

160

180FSMC

Generation

Mea

n F

itnes

s

36

0 5 10 15 20 25

0

5

10

15

Alti

tude

,m

Degraded Linaer Model

0 5 10 15 20 25

0

5

10

15Nonlinear Model

time,s

Alti

tude

,m

saic=0 deg

saic=10 deg

37

0 2 4 6 8 100

1

2

3

4

5

time,sel

evat

or d

efle

ctio

n,de

g

FSMC

1st compensator

2nd compensator

38

0 5 10 15 20 250

5

10

15

altit

ude,

m

1st Compensator

0 5 10 15 20 250

10

20

5

15

altit

ude,

m

2nd Compensator

0 5 10 15 20 250

5

10

15

time,s

altit

ude,

m

FSMC

k=2

k=0.5

39

0 1 2 3 4 5 6 7 8-1

-0.5

0

0.5relative velocity, m/s

0 1 2 3 4 5 6 7 8-5

0

5relative angle of attack,deg

0 1 2 3 4 5 6 7 8-20

0

20

40

time,s

pitch rate,deg/s

FSMC

1st compensator

2nd compensator