Attitude Controller Design of Mini-unmanned Aerial

8/9/2019 Attitude Controller Design of Mini-unmanned Aerial

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Published in IET Control Theory and Applications

Received on 27th April 2011

Revised on 26th October 2011

doi: 10.1049/iet-cta.2011.0240

ISSN 1751-8644

Attitude controller design of mini-unmanned aerialvehicles using fuzzy sliding-mode control degradedby white noise interferenceF.-K. Yeh

Department of Computer Science and Information Engineering, Chung Chou University of Science andTechnology,

Changhua 510,Taiwan

E-mail: fk [email protected]

Abstract: This work addresses an attitude controller design using a fuzzy sliding-mode controller, which is mainly composed of the sliding-mode controller and the fuzzy inference mechanism, for a mini-unmanned aerial vehicle (mini-UAV) with

propellers, to follow the predetermined trajectory. In this study, a fuzzy sliding-mode controller with a sliding surface vector is proposed, such that a simple fuzzy inference mechanism is adopted with centre of membership functions to estimate theupper bound of lumped uncertainty. Accordingly, the attitude trajectory tracking control using the proposed control strategy isillustrated while a mini-UAV is flying. Finally, the authors use the Lyapunov stability theory to achieve the stability analysisof the overall attitude tracking system degraded by white noise interference. Extensive simulation results are obtained tovalidate the effectiveness of the fuzzy sliding-mode controller.

Nomenclature

J moment of inertia matrix P roll rateQ pitch rate

R yaw rateS sliding surface vector V Lyapunov functiond disturbancel rolling momentm pitching momentn yawing momentt present timet reach time to reach sliding surfaceφ bank angleθ pitch angleψ heading

1 Introduction

An unmanned aerial vehicle (UAV) is an aircraft that flieswithout a human crew on board the aircraft. Many hobbyistsembark on developing UAVs for surveillance, detectionand destroying specific targets etc. This paper focuses onthe attitude tracking in the development of a situationalawareness system that allows for safe operation of UAVs.

For the reasons stated above, we know that the attitudecontrol of UAVs for the space development is essentialto successfully operate with higher level of autonomy and

to execute very complex military and civilian applications[1, 2].

A distributed algorithm is developed for task assignment,coordination and communication of multiple UAVs engagingmultiple targets to conceive an ad hoc routing algorithmfor synchronisation of target lists using a distributed computing topology [3]. Merino et al . [4] present acooperative perception system for multiple heterogeneousUAVs with different kinds of sensors: infrared and visual cameras and fire detectors to implement cooperative

perception techniques on automatic forest fire detectionand localisation. To develop guidance strategy and controldesign, a hand-launched small UAV has been addressed tooffer reliable aerial platform for ground reconnaissance [5].

To deal with the non-linear mini-UAV dynamics, weemploy the Eulerian representation [6, 7] to model thekinematic equations of rotational motion and the timederivative of angular velocities, so that the attitude trackingcontrol is applicable. To cope with the non-ideal factorssurrounding the mini-UAV of the external disturbances suchas wind and noise, robust feedback control shall be applied to ensure stability and system performance. Various controlstrategies have been proposed by many researchers in thelast few years to control mini-UAV dynamics and to designthe effective control laws such as: H-infinity control [8 – 10],which presented approaches to design compensators for attitude tracking using H-infinity output-feedback designtechniques, is employed to guarantee the robust stability;

linear matrix inequality (LMI) control algorithm [11],which presented a micro-helicopter controller design via atypical LMI approach, is proposed to compare the effects

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between the LMI approach and the sum of square approach;adaptive control method [12, 13], which proposed anadaptive strategy to enhance capabilities of the mini-UAVnavigation and the attitude angular velocity tracking control,respectively, is used to estimate unknown parameters.

A fuzzy gain scheduler [14] is introduced and the stabilitycondition is derived for being capable of vertical take-off and landing. A fuzzy logic based on closed-loop strapdown

attitude algorithm [15] is utilised for online scheduling of the parameters for the filtering estimator. Backstepping methods[16, 17] are presented for tracking controller design of the

path following control and for velocity control of a mini-UAV. The PID control [18] is also utilised to validate in bothhardware-in-the-loop simulations and actual experiments,and a hybrid PID velocity control method based on a singleneuron [19] achieves stable attitude in hover and improvesthe stability of the attitude control system. A wind tunnelexperiment for a generic unmanned combat aerial vehicleis conducted at a variety of angles of attack up to 70◦ and with various frequencies of pitch oscillation to investigatethe non-linear aerodynamic characteristics [20].

We recognised that the sliding-mode control is an

effective method for mini-UAVs with propellers to eliminatesystem uncertainties. Despite the popularity of such a controltechnique, it is, however, well known that the chattering

problem is worthy of more attention for the sake of practicaldeployment. Taking into consideration the aforementioned reason, a guide to sliding-mode control for practicalimplementation has been proposed by Young et al . [21]. Thechattering phenomenon can be solved by using the saturationfunction [22], which is a non-continuous solution, or thehyperbolic tangent function [23], which is a continuoussolution, respectively, to replace the sign function. Levant[24] proposed a super-twisting algorithm, which is robustto disturbances, using second-order sliding-mode control.

Yeh [25] presents a spacecraft attitude manoeuvring controlsystem using adaptive sliding-mode control to completestability and performance analysis. A feedback linearisation-

based controller with a high-order sliding-mode observer is proposed to be applied to a quadrotor UAV [26]. Galzi and Shtessel [27] proposed formations of mini-UAV using high-order sliding-mode controllers with robust and continuouscharacteristics to achieve leader/followers collision-free

path-tracking effect in the presence of unknown bounded disturbances. A sliding-mode controller incorporated with

backstepping approach is used by Bouadi et al. [28] for the synthesis of tracking errors to ensure Lyapunov stabilityand for the handling of all system non-linearities and desired tracking trajectories. Considering a famous non-

linear control method, the constructive non-linear controlof descriptive stability and optimality concepts has been

proposed by Kokotovic and Arcak [29] to be applied toa feedback passivation design for a turbocharged dieselengine.

Synergetic control theory and method of analytical designof aggregated regulators have been proposed by Kolesnikovand Kuzmenko [30] for compensating the external harmonicdisturbances to achieve the asymptotic stability of theclosed-loop power system, and by Kolesnikov [31] for finding of common objective laws of control processes ina forced dynamics system, as well as by Kondratiev et al .[32] for a DC–DC buck converter with constant power load.

In this paper, we investigate the fuzzy sliding-modecontroller design for mini-UAVs with propellers, using thesliding-mode control to eliminate the lumped uncertaintyand the fuzzy inference mechanism to estimate the upper

bound of lumped uncertainty. The paper is organised asfollows. In Section 2, vehicle attitude dynamic modelis established to represent the dynamic equations of angular displacements and angular velocities of mini-UAVsequipped with propellers. In Section 3, we propose thesliding-mode attitude controller aiming for tracking the

predetermined attitude trajectory. For tracking realisation,simulation results incorporating the so-called Euler-based

attitude control are developed in Section 4. To demonstratethe superior property of the proposed attitude controller,numerical simulations are provided in that section. Finally,conclusions are drawn in Section 5.

2 Vehicle attitude dynamic model

Attitude dynamic is the modelling of the changing positionand orientation of vehicles, because of external forces actingon the body. In general, vehicle actuators are used to achievea desired attitude, whereas attitude determination is theutilisation of vehicle sensors to ascertain the current vehicleattitude. The attitude dynamics of a mini-UAV with two

mains and two minor propellers are considered, referringto Fig. 1. To establish the attitude dynamic model of thevehicle, the flat-earth assumption is utilised because it isoperated over a small local region on earth. The attitudedynamic model of the vehicle is mapped by a set of threehighly non-linear differential equations [6, 7], namely

l = J x ˙ P − J zx( ˙ R + PQ) − ( J y − J z )QR (1)

m = J y Q − J zx( R2 − P 2) − ( J z − J x) RP (2)

n = J z ˙ R − J zx( ˙ P − QR) − ( J x − J y) PQ (3)

where l , m and n are the rolling, pitching and yawing

moments, respectively, P (t ), Q(t ) and R(t ) are the roll, pitchand yaw rates, respectively. The moment of inertia matrix

J b is symmetric, positive definite, and defined by

J b =

J x J xy J xz

J yx J y J yz

J zx J zy J z

(4)

Since xz -plane of the mini-UAV is the plane of symmetry,the products of inertia J xy and J yz are thus vanished.

Fig.1 Scheme of UAV

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Rearranging (1)–(3) and adding external disturbance models,yields

˙ P =1

J x J z − J 2 xz

{ J z [l + ( J y − J z )QR]

+ [n + ( J x − J y + J z ) PQ − J xz QR] J xz } + d 1 (5)

Q =1

J y[m + ( J z − J x) RP + J zx( R2 − P 2)] + d 2 (6)

˙ R =1

J x J z − J 2 xz

{ J x[n + ( J x − J y) PQ]

+ [l − ( J x − J y + J z ) RQ + J xz PQ] J xz } + d 3 (7)

For simplicity, if all components of the mini-UAV are properly allocated, then the yz -plane may be designed to be nearly a plane of symmetry, so that the product of inertia J xz is nearly vanished. This design constraint mayreduce the coupling effects of rigid body dynamics and thusdecrease the burden of the attitude dynamic control system.The schematic configuration of the vehicle is presented in

Fig. 1. The vehicle has two mains and two minor propellers.Therefore the attitude dynamic model can be yielded as

˙ P =( J y − J z )QR

J x+

l

J x+ d 1 (8)

Q =( J z − J x) RP

J y+

m

J y+ d 2 (9)

˙ R =( J x − J y) PQ

J z

+n

J z

+ d 3 (10)

The direction and thrust of the minor propellers can bemodulated to produce desired external moments, that is, l ,m and n. We assume that the thrust of each propeller isdirectly controlled, that is, we do not consider the dynamicsof rotors and propellers. The relationship between the body

rate ( P , Q and R) and the Euler rates ( ψ , θ and φ) is givenas [6, 7]

φ = P + tan θ (Q sin φ + R cos φ) (11)

θ = Q cos φ − R sin φ (12)

ψ = (Q sin φ + R cos φ)/ cos θ (13)

Equations (8)–(13) constitute the equations of motion of thevehicle for attitude dynamics.

3 Fuzzy sliding-mode attitude controller

Considering the presence of lumped uncertainty, mainlyconsisting of model uncertainties, parameter variations and disturbances, we recognised that the sliding-mode controlis an effectively robust controller for various applications.In this paper, we design the fuzzy sliding-mode controller not only to eliminate the lumped uncertainty but also toestimate the upper bound of lumped uncertainty with a fuzzyinference mechanism for a mini-UAV during the wholeflying course.

In general, there are two phases, which are reaching

and sliding conditions, to be suitably chosen to satisfy theconvergence requirement for the proposed controller design.If the sliding condition is verified to satisfy the requirementof sliding-mode controller, then the exponential stability of

the attitude control for a mini-UAV can be guaranteed aslong as the sliding surface vector is appropriately chosen.

Remark 1: First, to choose the sliding surface vector suchthat the sliding condition will be satisfied and hence theorigin of tracking error’s dynamics for the sliding surfacevector is exponentially stable, that is

S = [S 1 S 2 S 3]T

= [ ˙φ + λ1 φ ˙θ + λ1 θ ˙ψ + λ1 ψ]T

(14)

where λ1, λ2 and λ3 are positive-definite values. Denoteφ = φ − φ0, θ = θ − θ 0 and ψ = ψ − ψ0 to be the angular displacement errors, whereas φ, θ , ψ and φ0, θ 0, ψ0 arethe current angular displacements and the constant angular displacements with respect to the x-, y- and z -axis of themini-UAV body frame, respectively. From the sliding-modecontrol, once the reaching condition is satisfied, the systemis eventually forced to stay on the sliding surface vector. Bysliding-mode control, if the reaching condition is satisfied,then the sliding surface vector will be forced to be the zerovector, that is, S

3×1 = 0

3×1. It can be proved that the angular

displacement errors φ, θ and ψ will be driven to be zeroinherently. When the system is constrained in the sliding-

mode dynamics and so will the error derivatives ˙φ,

˙θ and

˙ψ of the φ, θ and ψ being driven to be zero. For the

above reason, the origin ( φ, ˙φ, θ ,

˙θ , ψ ,

˙ψ) = 01×6 of tracking

error’s dynamics of the ideal system can be verified to beexponentially stable.

The upper bound of the lumped uncertainty, whichincludes external disturbances and internal perturbations, for the sliding-mode control needs to be determined before thecontroller is running. In general, the upper bound of the

lumped uncertainty is difficult to be obtained for sliding-mode control in advance by computing, but always by themethod of trial and error. In this section, a fuzzy sliding-mode controller is proposed, in which a fuzzy inferencemechanism is used to estimate the upper bound of lumped uncertainty. We recognised that the prior expert knowledgeof the fuzzy inference mechanism, which can be used toestimate the upper bound of the lumped uncertainty, isavailable effectively.

The control block diagram of a fuzzy sliding-modecontroller is depicted as in Fig. 2. Based on fuzzy set theory,the associated fuzzy sets involved in the fuzzy control rulesare defined and listed as follows:

P positiveZ zero

N negativePH positive hugePB positive bigPM positive mediumPS positive smallZE zero

NS negative small NM negative medium NB negative big NH negative huge

Here, their universe of discourse is all assigned to be[−8, 8]. The membership functions for the fuzzy setscorresponding to sliding surface vector S , their derivatives





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Fig.2 Control block diagram of a fuzzy sliding-mode controller

Fig.3 Membership functions of fuzzy sets

S i, i = 1, 2, 3 and the upper bound of the lumped uncertainties τ i, i = 1, 2, 3 are defined in Fig. 3.

Thus, the three fuzzy subsets, N, Z and P for slidingsurface vector are defined in terms of S i and S i, i = 1, 2, 3,respectively. The fuzzy inference mechanism contains onlynine rules. The resulting fuzzy inference rules are as follows:

Rule 1: if S i is P and S i is P then τ i is PH;Rule 2: if S i is P and S i is Z then τ i is PB;

Rule 3: if S i is P and S i is N then τ i is PM;

Rule 4: if S i is Z and S i is P then τ i is PS;

Rule 5: if S i is Z and S i is Z then τ i is ZE;

Rule 6: if S i is Z and S i is N then τ i is NS;

Rule 7: if S i is N and S i is P then τ i is NM;

Rule 8: if S i is N and S i is Z then τ i is NB;

Rule 9: if S i is N and S i is N then τ i is NH.

Rule 1 is the condition that if the sliding variable S i and the derivative S i of S i are both positive then the upper bound

of the lumped uncertainty τ i is positive and huge for theoccurrence of sliding condition. Moreover, Rule 5 is thecondition that if the sliding variable S i and the derivativeS i of S i are both zero then the upper bound of the lumped

uncertainty τ i is zero for the occurrence of sliding condition.In turn, the same analysis can be made to explain the other fuzzy rules.

The fuzzy outputs τ i, i = 1, 2, 3, can be calculated by thecentre of area (COA) defuzzification as

τ i =

9

j =1 ωij cij 9

j =1 ωij

=

[ci1 · · · ci9]

ωi1

...ωi9

9

j =1 ωij

= |cT

i ωi|,

i = 1, 2, 3 (15)

where cTi = [ci1 · · · ci9] is an adjustable parameter vector,

ci1 through ci9 are the centre of the membership functions of τ i, and ωT

i = ([ωi1 · · · ωi9]/9

j =1 ωij ) is a firing strength

vector. Here, the absolute value of cTi ωi in (15) is used

to satisfy the requirement of that the upper bound of uncertainties is greater than or equal to zero, that is, τ i ≥ 0.The following theorem is introduced to enable the fuzzy

sliding-mode controller such that the reaching condition is





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satisfied. By Lyapunov stability theory, the fuzzy sliding-mode attitude controller of the mini-UAV can be proved as an exponentially stable system. The sufficient conditionsfor successful stability effect are stated in the followingtheorem.

Theorem 1: Fuzzy sliding-mode attitude controller: Let thestate space dynamic model be given by (8)–(13), and if thecontrol input moments l , m and n are proposed as

l = −( J y − J z )QR + J x ˙ P

m = −( J z − J x) RP + J y Q

n = −( J x − J y) PQ + J z ˙ R

(16)

where ˙ P

Q˙ R

= C −1

− A

φ

θ

ψ

− B

(17)

and (see equation at the bottom of the page)

where τ 1, τ 2 and τ 3 are small positive values, which arecomputed by (15), and S 1, S 2 and S 3 can be chosen usinga practical technique to avoid the singularity for computingthe inversion of the square matrix C . Denote

sign (S ) =

1 S > 0

0 S = 0

−1 S < 0

to be the sign function. Then the exponential stability and

robustness of the fuzzy sliding-mode attitude control systemcan be achieved.

Proof: To achieve the exponential stability and convergenceof the mini-UAV attitude tracking system designed by thefuzzy sliding-mode control, the non-linear attitude controller for the mini-UAV is developed such that the reachingcondition is satisfied. The main goal of this controller isto eliminate the lumped uncertainty of the attitude trackingsystem in the rotational directions, that is, the current angular displacements φ, θ and ψ will reach the constant angular displacements φ0, θ 0 and ψ0 in the body coordinate astime approaches to infinity. The system dynamics are then

constrained by the following derivations, referring to (14), as

S = [S 1 S 2 S 3]T = [ ¨φ + λ1

˙φ

¨θ + λ1

˙θ

¨ψ + λ1

˙ψ]T

(18)

In order to analyse the stability of the mini-UAV attitudetracking system, we define the sliding surface vector as (14).

Now we employ Lyapunov’s stability theory to define theLyapunov function as

V =1

2S 21 +

1

2S 22 +

1

2S 23 (19)

where S = [S 1 S 2 S 3]T. Differentiating (19), we have

V = S 1 S 1 + S 2 S 2 + S 3 S 3 (20)

Substitute (8)–(13) and (18) into (20), and let the control

input moments l , m and n be given by (16) a n d (17);therefore (20) can be further proposed as

V = −(S 1 + τ 1)sign (S 1) − (S 2 + τ 2)sign (S 2)

− (S 3 + τ 3)sign (S 3)

≤ −|S 1| − |S 2| − |S 3| (21)

where τ 1, τ 2 and τ 3 are small positive values.To avoid the singularity for computing the inversion of

the square matrix C , the determinant of matrix C can becomputed as

|C | = S 1

S 2

S 3 cos θ (22)

We proposed a practical technique that if any element of sliding-surface vector is equal to zero, that is, S i = 0, i =1, 2, 3, then the sliding surface vector S = [S 1 S 2 S 3]T

can be substituted by S = [S 1 S 2 S 3]T, where S 1, S 2 and S 3 can be chosen to be small positive values. From (22),to avoid the singularity for computing the inversion of thesquare matrix C , the pitch angle cannot work for that the

pitch angle is equal to ±90◦. For the value of cos θ wecan constrain the angular tracking in a constraint angular displacement, that is, −90◦ < θ < 90◦. So that the value of cos θ is positive. Therefore, the singularity for computingthe inversion of the square matrix C can be avoided.

A =

− RS 1 tan θ sin φ S 1(Q sin θ sec2 θ + R cos φ sec2 θ + Q sin θ ) 0

−S 2(Q sin φ + R cos φ) 0 0

S 3(Q cos φ − R sin φ)

cos θ

S 3 sin θ(Q sin φ + R cos φ)

cos2 θ 0

B =

S 1(λ1 P + λ1 tan θ (Q sin θ + R cos φ)) + (S 1 + S 1τ 1)sign(S 1)

S 2λ2(Q cos φ − R sin φ) + (S 2 + S 2τ 2)sign(S 2)

S 3λ3(Q sin φ + R cos φ)

cos θ + (S 3 + S 3τ 3)sign(S 3)

C =S 1 S 1 sin θ tan θ S 1 cos φ tan θ

0 S 2 cos φ −S

2 sin φ

0 S 3 sin φ cos θ S 3 cos φ cos θ





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From the derivation in (19) and (21), we have the outsideof the scalar S 1, S 2 and S 3 as

1

2

d

d t S 2i ≤ −|S i|, i = 1, 2, 3 (23)

Integrating the above equations for S i(t = 0) > 0 and S i(t =0) < 0, i = 1, 2, 3, between t = 0 to t = t reach leads to

S i(t = t reach) − S i(t = 0) = 0 − S i(t = 0) ≤ −(t reach − 0)

(24)

and thus

t reach ≤ |S i(t = 0)| (25)

Therefore the sliding variables S i, i = 1, 2, 3 will beconverged to zero in a finite time. Finally, the main goalof achieving exponential stability and robustness of thesliding-mode attitude tracking system can be guaranteed completely.

Remark 2: However, because that the non-ideality in the

practical implementation of the sign function sign(S ) exists,the control input moments l , m and n in (16) and (17) alwayssuffers from the chattering problem. To alleviate such anundesirable phenomenon, the sign function can be simplyreplaced by the hyperbolic tangent function. The system isnow no longer forced to stay on the sliding surface vector

but is constrained within the boundary layer |S i| ≤ ε , i = 1,2, 3, where ε is a small positive value.

To alleviate the chattering phenomenon, the hyperbolictangent function may be used to the control input of theattitude tracking system with the sliding-mode controller.Consequently, the term τ i sign (S i), i = 1, 2, 3, in (17) can

be replaced by

τ i tanh(5S i/ε), i = 1, 2, 3 (26)

where ε is a small constant, and tanh( z ) = {(e z − e− z )/(e z +

e− z )}.

Therefore the matrix B in (17) for the control inputmoments can be re-expressed as (see (27))

4 Simulations

To validate the proposed attitude tracking problem, the fuzzysliding-mode attitude controller for a mini-UAV is designed.

To demonstrate the performance and effectiveness of the proposed controller, the verified simulations are given. Allsimulation results will be shown in the following contexts.

For simulation the initial conditions of angular displacements and angular velocities are set as [φ(0) θ (0)ψ(0)]T = [0 0 0]T and [ P (0) Q(0) R(0)]T = [0 00]T, respectively. Furthermore, the desired values of angular displacements [φ0 θ 0 ψ0]T = [0.3 0.7 1.2]T and thetime derivatives of the initial current angular displacements[ φ(0) θ (0) ψ(0)]T = [0 0 0]T are also given. The

moments of inertia of the mini-UAV are set as J x =

0.0144 kgm, J y = 0.0254 kgm and J z = 0.0312 kgm. In our simulation, some positive-definite values of λi = 5, i = 1,2, 3 [see (14)] for the sliding-surface vector and the upper

bound of lumped uncertainty τ i, i = 1, 2, 3 [see (15)] for eliminating the disturbances as well as S i = 0.0001, i = 1,2, 3 [see (22)] for avoiding the singularity for computingthe inversion of the square matrix C , are all adopted.

The appealing effect of the fuzzy sliding-mode attitudecontrol presented is given in Fig. 4, which shows thesuccessful tracking performance of the present attitude thatthe current angular displacements φ, θ and ψ are reached to the constant angular displacements φ0 = 0.3, θ 0 = 0.7 and ψ0 = 1.2 rad in the body coordinate. From Fig. 4, the attitudetracking effect is fulfilled almost totally after 6 s. This is toshow the feasibility of the conclusion from (21) and to showwell the results of attitude tracking of the mini-UAV. Here,the robustness and effectiveness can also be demonstrated

by the results shown in Fig. 5 for the angular velocitiesof the attitude control system. The sliding surface vector isshown in Fig. 6. From Fig. 6, the successful convergence of the sliding variables can be confirmed and that can also be

verified by (14) and (21). The control input moment of fuzzy

Fig.4 Angular displacements in the body coordinate frame

Fig.5 Angular velocities in the body coordinate frame

B =

S 1(λ1 P + λ1 tan θ (Q sin θ + R cos φ)) + S 1 tanh(5S 1/ε) + τ 1S 1 tanh(5S 1/ε)

S 2

λ2

(Q cos φ − R sin φ) + S 2 tanh(5S

2/ε) + τ

2S

2 tanh(5S

2/ε)

S 3λ3(Q sin φ + R cos φ)

cos θ + S 3 tanh(5S 3/ε) + τ 3S 3 tanh(5S 3/ε)

(27)





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Fig.6 Sliding surface vector of a fuzzy sliding-mode controller

Fig.7 Control input moments of a fuzzy sliding-mode controller

Fig.8 Control input moments of a sliding-mode controller

sliding-mode controller with fuzzy inference mechanism and of sliding-mode controller with constant upper bound of lumped uncertainty τ i = 0.05, i = 1, 2, 3 are shown inFigs. 7 and 8, respectively, to demonstrate the practicaleffect. From the comparison between Figs. 7 and 8, theoverall stability and control results of the mini-UAV attitudetracking system for the fuzzy sliding-mode controller withvariable upper bound of lumped uncertainty, which is shown

in Fig. 9 and their amplitudes are dependent on the slidingvariables S i and the derivatives S i, i = 1, 2, 3, are better than those for the sliding-mode controller with the constantupper bound. The disturbances d 1, d 2, and d 3, containing real

Fig.9 Upper bound of lumped uncertainty for a fuzzy

sliding-mode controller

Fig. 10 Real white Gaussian noises of power 1 dBW

white Gaussian noises of power 1 dBW, given in (5)–(7), areshown in Fig. 10.

5 Conclusions

Since the attitude control and system stability are keysfor aviation technology, we address the fuzzy sliding-modeattitude controller designs of a mini-UAV. For the non-linear attitude controller design, a fuzzy sliding-mode controller is

investigated in which a simple fuzzy inference mechanismis used to estimate the upper bound of lumped uncertainty.We prove the exponential stability in the conditions of those external disturbances and internal perturbations aremerged together through the aid of the Lyapunov stabilityanalysis. Extensive simulations have been adopted to verifythe feasibility of the fuzzy sliding-mode attitude controllers.The system performance and its stability can also bedemonstrated by use of the aforementioned theoreticalderivations and the realistic simulations.

6 Acknowledgment

The author would like to thank the National Science Councilof the Republic of China, Taiwan, for financially supportingthis research under Contract No. NSC 98-2221-E-235-008.





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Attitude Controller Design of Mini-unmanned Aerial

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