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applying NN to model unknown functions in dynamicsystems has been demonstrated in several studies [7–12]

As such, several flight control approaches using NN havebeen proposed. Among of them, approximate dynamicinversion with augmented NN was proposed to handleunmodelled dynamics in [13– 15], whereas neural dynamic

programming was shown to be effective for tracking andtrimming control of helicopters in [16]. During theadaptive trajectory control of autonomous helicopter in [17]and [18], the method of pseudocontrol hedging was usedto protect the adaptation process from actuator limits anddynamics. In [19], MIMO output feedback adaptive NNcontrol was proposed for an autonomous scale modelhelicopter mounted in a 2-degree-of-freedom (2DOF)platform. In [20], robust adaptive NN based on the mean

value theorem and the implicit function theorem wasproposed to handle the nonaffine nonlinearities in thehelicopter dynamics without the construction of the

dynamic inversion.

In this paper, motivated by [6], where the exact modeldynamics are known, we consider the altitude and yaw angletracking for a scale model helicopter mounted on anexperimental platform in the presence of model uncertainties,

which may be caused by unmodelled dynamics, sensor errorsor aerodynamical disturbances from the environment.Compared with the model-based control used in [6],approximation-based control using NN, proposed in thispaper, can accommodate the presence of model uncertainties,reduce the dependence on accurate model building, and thus,lead to the tracking performance improvement.

2 Problem formulation andpreliminaries

In the following study, ~ () ¼ ^() (), let k k denote the 2-norm, k kF denote the Frobenius norm and j j1 denote1-norm, i.e. given A ¼ [a ij ] [ R mn, k A k2

F ¼ tr{ A T A } ¼Pi , j a 2i , j , and j A j1 ¼

Pi , j ja i , j j. The following definition

and technical lemma are required in the subsequent controldesign and stability analysis.

Definition 1 [11] : The solution X (t ) is semiglobally uniformly ultimately bounded (SGUUB) if, for any compact set V0 and all X (t 0) [ V0, there exists an m . 0and T (m, X (t 0)) such that k X (t )k m for all t t 0 þ T .

Lemma 1: For a , b [ R þ, the following inequality holds

ab

a þ b a (1)

2.1 Helicopter dynamics

In this paper, we consider a VARIO scale model helicopter [6] which is mounted on an experimental platform as shown inFig. 1, where the xyz and x 1 y 1z1 reference systems represent

an inertial frame and a body fixed one, respectively. Inaddition, _f is the yaw rate and _g is the main rotor angular

velocity. The counterbalance weight compensates for the weight of the vertical column of the platform. Thehelicopter dynamics is described by Lagrangian formulationin the following [6]

D (q )€q þ C (q , _q )_q þ F (_q ) þ G (q ) þ D(q , _q ) ¼ B (_q )t (2)

where q , _q and €q are referred as the vectors of generalisedcoordinates, generalised velocities and generalisedaccelerations, respectively. In particular, q ¼ [q 1, q 2, q 3] T ¼

[z, f , g ] T with z as the attitude (z . 0 downwards), f asthe yaw angle and g as the main rotor azimuth angle;

_q ¼ [_q 1, _q 2, _q 3] T ¼ [_z, _f , _g ] T with _z as the vertical velocity,_f as the yaw rate and _g as the main rotor angular velocity;

€q ¼ [€q 1, €q 2, €q 3] T ¼ [€z, €f , €g ] T with €z as the vertical

acceleration €f as the yaw acceleration and €g as the main

rotor angular acceleration; D (q ) [ R 33 is the inertia matrix;C (q , _q )_q [ R 31 is the vector of Coriolis and centrifugalforces; F (_q ) [ R 31 is the vector of friction forces;G (q ) [ R 31 is the vector of gravitational forces;D(q , _q ) [ R 31 is the vector of the model uncertainties,

which may be caused by unmodelled dynamics, sensor errorsor aerodynamical disturbances from the environment;B (_q ) [ R 32 is the matrix of control coefficients; and thecontrol inputs t ¼ [t 1, t 2] T [ R 21 are the main and tailrotor collectives (swash plate displacements), respectively. By exploiting the physical properties of the helicopter, e.g. how the control inputs are distributed to the helicopter dynamics,

or the coupling relationship between the states, better performance can be achieved. To this end, we assume partialknowledge of the structure of the dynamics [6], although

Figure 1 Helicopter-platform [6]

942 IET Control Theory Appl., 2009, Vol. 3, Iss. 7, pp. 941–956

& The Institution of Engineering and Technology 2009 doi: 10.1049/iet-cta.2008.0103

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The ideal weight vector W is defined as the value of W that minimises j1(Z )j for all Z [ VZ , R m, i.e.

W ¼ argminW

{ supZ [VZ

j f (Z ) W TS (Z )j}

In general, the ideal weights W are unknown and need to beestimated in control design. Let W be the estimates of W ,and the weight estimation errors ~ W ¼ W W .

2.2.2 Function approximation using MNN: In thispaper, we also consider nonlinearly parameterised MNN,

which is used to approximated the continuous function f (Z ) : R m ! R as follows

f (Z ) ¼ W TS (V TZ ) þ 1(Z )

where the vector Z ¼ [z1, z2, :::, zm, 1] T[ VZ , R mþ1 are

the input variables to the NNs; S () [ R l is a vector of knowncontinuous basis functions, with l denoting the number of neural nodes; W [ R l and V [ R (mþ1)l are adaptable

weights; and 1(Z ) is the approximation error which isbounded over the compact set VZ , i :e : j1(Z )j 1,8Z [ VZ where 1 . 0 is an unknown constant. According to the universal approximation property [21], MNN cansmoothly approximate any continuous function f (Z ) over a compact set VZ , R mþ1 to arbitrary any degree of accuracy as that

f (Z ) ¼ W TS (V TZ ) þ 1(Z ), 8Z [ VZ , R mþ1

where W and V are the ideal constant weights, and 1(Z ) is

the approximation error for the special case where W ¼ W

and V ¼ V . The ideal weights W and V are defined asthe values of W and V that minimise j1(Z )j for allZ [ VZ , R mþ1, i.e.

(W , V ) :¼ arg min(W ,V )

{ supZ [VZ

j f (Z ) W TS (V TZ )j}

Assumption 7: On the compact set VZ , the ideal NN weights W , V are bounded by

kW k wm, kV kF v m

In general, the ideal weights W and V are unknown andneed to be estimated in control design. Let W and V bethe estimates of W and V , respectively, and the weight estimation errors ~ W ¼ W W and ~ V ¼ V V .

Lemma 2 [11] : Using f mnn ¼ W TS ( V TZ ) to approximatethe ideal function f (Z ), its approximation error can beexpressed as

^

W

T

S ( ^

V

T

Z ) W

T

S (V

T

Z )

¼ ~ W T

(S S 0V TZ ) þ W TS 0 ~ V T

Z þ d u

where S ¼ S ( V TZ ), S 0 ¼ diag {S 01, S 02, . . . , S 0l } with

S 0i ¼ S 0(v Ti Z ) ¼d[s (za )]

dza

jza ¼v Ti Z

and the residual term d u is bounded by

jd u j kV kF kZ W TS 0kF þ kW kkS 0V TZ k þ jW j1

Throughout this paper, we employ sigmoidal functions asbasis functions for the MNN, which are defined by

s i (za ) ¼1

1 þ emza , i ¼ 1, 2, . . . , l (8)

where m . 0 is a design constant.

3 Control design

Motivated by the previous work on model-based control of helicopters [6], we will design adaptive neural control toaccommodate the presence of uncertainties in the dynamics(2), appearing in the functions D (q ), C (q , _q ), F (_q ), G (q ),D(q , _q ) and B (_q ). After some simple manipulations on (2)and (3), we can obtain three subsystems: q 1 subsystem (9),q 2 subsystem (10) and q 3 subsystem (11) as follows

d 11 €q 1 þ f 1(_q 3) þ g 1 þ D1(q , _q ) ¼ b 11(_q 3)t 1 (9)

d 22(q 3)d 33 d

2

23d 33

€q 2 þ c 22(q 3, _q 3)_q 2 þ c 23(q 3, _q 2)_q 3

þ D2(q , _q ) þd 23

d 33

(b 31(_q 3)t 1 c 32(q 3, _q 2)_q 2

f 3(_q ) g 3 D3(q , _q )) ¼ b 22(_q 3)t 2 (10)

d 22(q 3)d 33 d 223

d 22(q 3) €q 3 þ c 32(q 3, _q 2)_q 2 þ f 3(_q 3) þ g 3

þ D3(q , _q ) þd 23

d 22(q 3)(b 22(_q 3)t 2 c 22(q 3, _q 3)_q 2

c 23(q 3, _q 2)_q 3 D2(q , _q )) ¼ b 31(_q 3)t 1 (11)

In the following, we will analyse and design control for eachsubsystem. For clarity, define the tracking errors and thefiltered tracking errors as

e i ¼ q i q id , r i ¼ _e i þ li e i (12)

where li is a positive number, i ¼ 1, 2. Then, theboundedness of r i guarantees the boundedness of e i and _e i [22–25]. To study the stability of e i and _e i , we only needto study the properties of r i . In addition, the following computable signals are defined

_q ir ¼ _q id li e i , €q ir ¼ €q id li _e i



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3.1 RBFNN-based control

In this section, we will investigate the RBFNN-based controldesign by Lyapunov synthesis to achieve the control objective.Regarding the obtained three subsystems (9) –(11), our control design consists of three steps: first, we will design

control t 1 based on the q 1 subsystem (9) second, design t 2based on the q 2 subsystem (10) and t 1; finally, analyse thestability of internal dynamics of q 3 subsystem (11).

3.1.1 q1 subsystem: Since _q 1 ¼ _q 1r þ r 1, €q 1 ¼ €q 1r þ _r 1,(9) becomes

d 11 _r 1 ¼ b 11(_q 3)t 1 f S 1,1 (13)

where

f S 1,1 ¼ d 11 €q 1r þ f 1(_q 3) þ g 1 þ D1(q , _q ) (14)

is an unknown continuous function, which is approximatedby RBFNN to arbitrary any accuracy as

f S 1,1 ¼ W T1 S 1(Z 1) þ 11(Z 1) (15)

where the input vector Z 1 ¼ [q 1, _q 1, q 2, _q 2, q 3, _q 3,_q 1d , €q 1d ]

T[ VZ 1 , R 8; 11(Z 1) is the approximation error

satisfying j11(Z 1)j 11, where 11 is a positive constant;W 1 are ideal constant weights satisfying kW 1 k w1m,

where w1m is a positive constant; and S 1(Z 1) are the basis

functions. By using ^

W 1 to approximate W

1 , the error between the actual and the ideal RBFNN can beexpressed as

W 1 TS 1(Z 1) W T

1 S 1(Z 1) ¼ ~ W 1 T

S 1(Z 1) (16)

where ~ W 1 ¼ W 1 W 1 .

Consider the following Lyapunov function candidate

V 1 ¼1

2d 11r 21 þ

1

2~ W 1

TG

11

~ W 1 (17)

The time derivative of (17) along (13) and (15) is given by

_V 1 ¼ d 11r 1 _r 1 þ ~ W 1 TG

11

_~ W 1

¼ r 1[b 11(_q 3)t 1 W T1 S 1(Z 1) 11(Z 1)] þ ~ W 1

TG

11

_~ W 1

(18)

As W 1 is a constant vector, we know that _~ W 1 ¼ _W 1.

Therefore (18) becomes

_V 1 ¼ r 1[b 11(_q 3)t 1 W T1 S 1(Z 1) 11(Z 1)] þ ~ W 1 TG1

1 _W 1

(19)

Consider the following RBFNN based control law andRBFNN weight adaptation law

t 1 ¼ k1r 1 r 1( W 1

TS 1(Z 1))2

b 11(jr 1 W 1 TS 1(Z 1)j þ d 1)

(20)

_W 1 ¼ G1[S 1(Z 1)r 1 þ s 1 W 1] (21)

where k1 . 0, d 1 . 0, G1 ¼ G T1 . 0, and s 1 . 0.

Remark 4: The above s -modification adaptation law (21)can be replaced by e -modification adaptation law like

_W 1 ¼ G1[S 1(Z 1)r 1 þ s 1jr 1j W 1] easily. The control designbased on s -modification adaptation law in this paper canbe extended to the case based on e -modification adaptationlaw without any difficulty.

Substituting (20) and (21) into (19), we have

_V 1 ¼ k1b 11(_q 3)r 21 b 11(_q 3)

b 11

r 21 ( W 1 TS 1(Z 1))2

jr 1 W 1 T

S 1(Z 1)j þ d 1

r 1W T1

S 1(Z 1) r 111(Z 1) r 1 ~ W 1 T

S 1(Z 1) s 1 ~ W 1

TW 1

(22)

According to Assumption 3 and (16), we can rewrite (22) as

_V 1 k1b 11r 21 r 21 ( W 1

TS 1(Z 1))2

jr 1 W 1 T

S 1(Z 1)j þ d 1

r 1 W 1 TS 1(Z 1) r 111(Z 1) s 1

~ W 1 TW 1

k1b 11r 21 r 21 ( W 1

TS 1(Z 1))2

jr 1 W 1 T

S 1(Z 1)j þ d 1

þ jr 1 W 1 TS 1(Z 1)j þ jr 1k11(Z 1)j s 1

~ W 1 T

W 1 (23)

Noting that

r 21 ( W 1

TS 1(Z 1))2

jr 1 W 1 T

S 1(Z 1)j þd 1

þ jr 1 W 1 TS 1(Z 1)j ¼

jr 1 W 1 TS 1(Z 1)jd 1

jr 1 W 1 T

S 1(Z 1)j þd 1

(24)

According to Lemma 1, we can obtain from (24) that

r 21 ( W 1

TS 1(Z 1))2

jr 1 W 1 T

S 1(Z 1)j þ d 1

þ jr 1 W 1 TS 1(Z 1)j d 1 (25)

By completion of squares and using Young’s inequality, thefollowing inequalities hold

s 1 ~ W

T

1 W 1

s 12

k ~ W 1k2

þs 12

kW 1k2 (26)

jr 1k11(Z 1)j r 212c 1

þc 11

21(Z 1)

2

r 212c 1

þc 1 12

1

2 (27)

IET Control Theory Appl., 2009, Vol. 3, Iss. 7, pp. 941 – 956 945

doi: 10.1049/iet-cta.2008.0103 & The Institution of Engineering and Technology 2009

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where c 1 is a positive constant. Substituting the aboveinequalities (25)–(27) into (23) leads to

_V 1 k1b 11 1

2c 1

r 21

s 12

k ~ W 1k2

þ d 1 þc 12

121 þ

s 12

w21m

12

d 112k1b 11 (1=c 1)

d 11

r 21 12

s 1

lmax(G11 )

~ W 1 TG

11

~ W 1 þ d 1 þc 12

121 þ

s 12

w21m

min 2k1b 11 (1=c 1)

d 11

, s 1

lmax(G11 )

( )

1

2d 11r 21 þ

1

2~ W 1

TG

11

~ W 1

þ d 1 þ

c 12

121 þ

s 12

w21m

l10V 1 þ m10 (28)

where l10 ¼ min (2k1b 11 1=c 1)=d 11, s 1=lmax(G11 )

, m10 ¼

d 1 þc 12 1

21 þ

s 12 w2

1m.

3.1.2 q2 subsystem: Similar to Section 3.1.1, since_q 2 ¼ _q 2r þ r 2, €q 2 ¼ €q 2r þ _r 2, (10) becomes

d 22(q 3)d 33 d 223

d 33

_r 2 þ c 22(q 3, _q 3)r 2 ¼ b 22(_q 3)t 2 f S 2,1 (29)

where

f S 2,1 ¼d 22(q 3)d 33 d 223

d 33

€q 2r þ c 22(q 3, _q 3)_q 2r þ c 23(q 3, _q 2)_q 3

þ D2(q , _q ) þd 23

d 33

(b 31(_q 3)t 1 c 32(q 3, _q 2)_q 2

f 3(_q 3) g 3 D3(q , _q ))

is an unknown function, which is approximated by RBFNNto arbitrary any accuracy as

f S 2,1 ¼ W T

2 S 2(Z 2) þ 12(Z 2) (30)

where the input vector Z 2 ¼ [t 1, q 1, _q 1, q 2, _q 2, q 3,

_q 3, q 2d , _q 2d , €q 2d ] T[ VZ 2

, R 10, 12(Z 2) is theapproximation error satisfying j12(Z 2)j 12, where 12 is anunknown positive constant; W

2 are unknown ideal constant weights satisfying kW 2 k w2m, where w2m is an unknownpositive constant; and S 2(Z 2) are the basis functions. By using W 2 to approximate W 2 , the error between the actualand the ideal RBFNN can be expressed as

W T

2 S

2(Z

2) W T

2 S

2(Z

2) ¼ ~ W

T

2S

2(Z

2) (31)

where ~ W 2 ¼ W 2 W 2 .

To analyse the closed-loop stability for the q 2 subsystem,let

V 2 ¼1

2

d 22(q 3)d 33 d 223

d 33

r 22 þ1

2~ W

T

2 G12

~ W 2 (32)

Lemma 3: The function V 2 (32) is positive definite anddecrescent, in the sense that there exist two time-invariant positive definite functions V 2(r 2, ~ W 2) and V 2(r 2, ~ W 2), suchthat

V 2(r 2, ~ W 2) V 2 V 2(r 2, ~ W 2)

Proof: Noting that the particular choice of V 2 in (32), a function of r 2, ~ W 2 and d 22(q 3), is to establish the stability for r 2 and ~ W 2 only; therefore, we regard d 22(q 3) a s a function of time. From Assumptions 1 and 4, we know that

0 , jd 22jd 33j d 223j

jd 33j, j d 22(q 3)d 33 d

223

d 33

j d 22jd 33j þ d

223

jd 33j

(33)

Therefore there also exist time-invariant positive definitefunctions V 2(r 2, ~ W 2) and V 2(r 2, ~ W 2), such that V 2(r 2, ~ W 2) V 2 V 2(r 2, ~ W 2), which implies that V 2 is also positivedefinite and decrescent, according to [25]. This completesthe proof. A

The time derivative of (32) is given as

_V 2 ¼ 12 _d 22(q 3)r 22 þ d 22(q 3)d 33 d

2

23d 33

r 2 _r 2 þ ~ W T2 G12 _~ W 2

(34)

According to Assumption 2, (34) becomes

_V 2 ¼ r 2d 22(q 3)d 33 d 223

d 33

_r 2 þ c 22(q 3, _q 3)r 2

" #þ ~ W

T

2 G12

_~ W 2

(35)

As W 2 is a constant vector, it is easy to obtain that

_~ W 2 ¼ _W 2 (36)

Substituting (29), (30) and (36) into (35), we have

_V 2 ¼ r 2 b 22(_q 3)t 2 W T 2 S 2(Z 2) 12(Z 2)

þ ~ W

T

2 G12

_W 2

(37)

Consider the following RBFNN-based control law andRBFNN weight adaption law

t 2 ¼ k2r 2 þr 2( W 2

TS 2(Z 2))2

b 22(jr 2 ^

W 2 T

S 2(Z 2)j þ d 2)

(38)

_W 2 ¼ G2[S 2(Z 2)r 2 þ s 2

W 2] (39)



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where k2 . 0, d 2 . 0,G2 ¼ G T2 . 0ands 2 . 0. Substituting

(38) and (39) into (37), we have

_V 2 ¼ k2b 22(_q 3)r 21 þb 22(_q 3)

b 22

r 22 ( W 2 TS 2(Z 2))2

jr 2 W 2 T

S 2(Z 2)jþd 2

r 2W T2 S 2(Z 2)

r 212(Z 2)r 2 ~ W 2 TS 2(Z 2)s 2

~ W 2 TW 2 (40)

According to Assumption 3 and (31), we can rewrite (40) as

_V 2 k2b 22r 22 r 22 ( W 2

TS 2(Z 2))2

jr 2 W 2 T

S 2(Z 2)j þd 2

r 2 W 2 TS 2(Z 2)r 212(Z 2) s 2

~ W 2 T

W 2

k2b 22r 22 r 22 ( W 2

TS 2(Z 2))2

jr 2 W 2 T

S 2(Z 2)j þd 2

þjr 2 W 2 TS 2(Z 2)j þjr 2k12(Z 2)j s 2

~ W 2 T

W 2 (41)

Similar to (25), we have

r 22 ( W 2

TS 2(Z 2))2

jr 2 W 2 T

S 2(Z 2)jþ d 2

þjr 2 W 2 TS 2(Z 2)j d 2 (42)


s 2 ~ W

T

2 W 2

s 2

2

k ~ W 2k2

þs 2

2

kW 2k2 (43)

jr 2k12(Z 2)j r 222c 2

þc 21

22(Z 2)

2

r 222c 2

þc 2 1

22

2 (44)

where c 2 is a positive constant. Substituting the aboveinequalities (42)–(44) into (41) leads to

_V 2 k2b 22 1

2c 2

r 22

s 22

k ~ W 2k2

þd 2 þc 22

122 þ

s 22

w22m

l20V 2 þm20 (45)

where l20 ¼ min{(2k2b 22 1=c 2)jd 33j=( d 22jd 33jþ d 223),s 2=

lmax(G12 )}, m20 ¼d 2 þ

c 22 1

22 þ

s 22 w2

2m.

3.1.3 q3 subsystem: Finally, using the designed controllaws (20) and (38), the q 3-subsystem (1) can be rewritten as

_h ¼ c (j , h , u ) (46)

where h ¼ [q 3, _q 3] T, j ¼ [q 1, q 2, _q 1, _q 2] T, u ¼ [t 1, t 2] T.

Then, the zero-dynamics can be addressed as [26]

_h ¼ c (0, h , u

(0, h )) (47)

where u ¼ [t 1, t 2] T.

Assumption 8 [26] : System (9) (10) (11) is hyperbolically minimum-phase, i.e. zero-dynamics (47) is exponentially stable. In addition, assume that the control input u isdesigned as a function of the states (j , h ) and the referencesignal satisfying Assumption 5, and the function f (j , h , u )is Lipschitz in j , i.e. there exist constants Lj and L f for

f (j , h , u ) such that

k f (j , h , u ) f (0, h , u h )k Lj kj k þ L f (48)

where u h ¼ u (0, h ).

Under Assumption 8, by the Converse Theorem of Lyapunov [27], there exists a Lyapunov function V 0(h ) whichsatisfies

g a kh k2

V 0(h ) g b kh k2 (49)

@V 0@h

f (0, h , u h ) la kh k2 (50)

k@V 0@h

k lb kh k (51)

where g a , g b , la and lb are positive constants.

Lemma 4 [26] : For the internal dynamics _h ¼ f (j , h , u )of the system, if Assumption 8 is satisfied, and the states j are bounded by a positive constant kj kmax, i.e.kj k kj kmax, then there exist positive constants Lh andT 0, such that

kh (t )k Lh , 8t > T 0 (52)

Proof: According to Assumption 8, there exists a Lyapunov function V 0(h ). Differentiating V 0(h ) along (9), (10), and(11) yields

_V 0(h ) ¼@V 0@h

f (j , h , u )

¼@V 0@h

f (0, h , u h ) þ@V 0@h

[ f (j , h , u ) f (0, h , u h )]

(53)

Noting (48)–(51), (53) can be written as

_V 0(h ) la kh k2

þ lb kh k(Lj kj k þ L f )

la kh k2

þ lb kh k(Lj kj kmax þ L f )

Therefore _V 0(h ) 0, whenever

kh k lb

la

(Lj kj kmax þ L f )

By letting Lh ¼ lb =r la (Lj kj kmax þ L f ), we conclude that there exists a positive constant T 0, such that (52) holds. A



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The following theorem shows the stability and controlperformance of the closed-loop system.

Theorem 1: Consider the closed-loop system consisting of the subsystems (9)–(11), the control laws (20), (38) and

adaptation laws (21), (39). Under Assumptions 1–8, theoverall closed-loop neural control system is SGUUB in thesense that all of the signals in the closed-loop system arebounded, and the tracking errors and neural weightsconverge to the following regions

je 1j je 1(0)j þ1

l1

ffiffiffiffiffiffiffiffi2m1

d 11

s , k W 1k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m1

lmin(G11 )

s þ w1m

je 2j je 2(0)j þ1

l2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2jd 33jm2

jd 22jd 33j d 223j

s ,

k W 2k


lmin(G12 )

s þ w2m

(54)

with

mi ¼mi 0

li 0

þ V i (0), mi 0 ¼ d i þ1

2 1

2i þ

s i 2

w2im, i ¼ 1, 2

l10 ¼ min{(2k1b 11 1=c 1)=d 11, s 1=lmax(G11 )}

l20 ¼ min{(2k2 1=c 2)jd 33j=( d 22jd 33j þ d 223), s 2=lmax(G12 )}

where e i (0) and V i (0) are initial values of e i (t ) and V i (t ),respectively.

Proof: Based on the previous analysis, the proof also proceedsby studying each subsystem in order. First, the closed-loopstability analysis of q 1 subsystem (9) with control t 1 (20) andadaptation law (21) is made by using Lyapunov synthesis.Second, similar closed-loop stability will be achieved on q 2subsystem (10) with t 2 (38) and adaptation law (39).Finally, the stability analysis of internal dynamics of q 3subsystem (11) is made based on the stability of the previoustwo subsystems.

q 1-subsystem: Solving the inequality (109), we have0 V 1(t ) m1 with m1 ¼ (m10=l10) þ V 1(0). Then, fromthe definition of V 1(t ) (17), we can obtain

jr 1j


d 11

s , k ~ W 1k


lmin(G11 )

s (55)

Since _e 1 ¼ l1e 1 þ r 1, solving this equation results in

e 1 ¼ el1t e 1(0) þ

ð t

0el1(t t )r 1 dt (56)

According to (55) and (56), we have

je 1j je 1(0)j þ1

l1


d 11

s (57)

Noting q 1 ¼ e 1 þ q 1d , W 1 ¼ ~ W 1 þ W 1 , kW 1 k w1m and Assumption 5, we obtain

jq 1j je 1j þ jq 1d j je 1(0)j þ1

l1


d 11

s þ jq 1d j [ L

1

k W 1k k ~ W 1k þ kW 1 k


lmin(G11 )

s þ w1m [ L

1

Since the control t 1 is a function of r 1 and W 1, its

boundedness is also guaranteed.

q 2-subsystem: Similar to the analysis of q 1 subsystem, we have

jr 2j


d 22jd 33j d 223

s

, k ~ W 2k


lmin(G12 )

s (58)

Furthermore, we obtain

je 2j je 2(0)j þ1

l2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2jd 33jm2

jd 22jd 33j d 223j

s jq 2j je 2j þ jq 2d j je 2(0)j

þ1

l2


jd 22jd 33j d 223j

s þ jq 2d j [ L

1

k W 2k k ~ W 2k þ kW 2 k


lmin(G12 )

s þ w2m [ L

1 (59)

and thus the boundedness of control t 2.

q 3-subsystem: From the previous stability analysis about the q 1subsystem and the q 2 subsystem, we know that q 1, q 2, _q 1, _q 2are bounded. Accordingly, j are bounded. According toLemma 5, we know that the internal dynamics is stable,i.e. h (q 3 and _q 3) are bounded. All the signals in the closed-loop system are bounded. This completes the proof. A

3.2 MNN-based control

Nonlinearly parameterised approximators, such as MNN, canbe linearised by Taylor series expansions,with the higher order terms being taken as part of the modelling error. Due to the

nonlinear parameterisation, the control design and stability analysis involving MNN are more complex than that basedon the linearly parameterised network RBFNN.



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3.2.1 q1 subsystem: Similar to the RBFNN case inSection 3.1.1, (9) is written as

d 11 _r 1 ¼ b 11(_q 3)t 1 f S 1,1 (60)

where the unknown continuous function

f S 1,1 ¼ d 11 €q 1r þ f 1(_q 3) þ g 1 þ D1(q , _q ) (61)

is approximated by MNN to arbitrary any accuracy as

f S 1,1 ¼ W T1 S 1(V T

1 Z 1) þ 11(Z 1) (62)

where the input vector Z 1 ¼ [q 1, _q 1, q 2, _q 2, q 3, _q 3, _q 1d ,€q 1d , 1] T [ VZ 1

, R 9; 11(Z 1) is the approximation error satisfying j11(Z 1)j 11, where 11 is a positive constant;W 1 and V 1 are unknown ideal constant weights satisfying

kW

1 k w1m, kV

1 kF v 1m, which are positive constants.By using W T1 S 1( V T1 Z 1) to approximate W T

1 S 1(V T1 Z 1), the

error between the actual and the ideal MNN can beexpressed as

W T1 S ( V T1 Z 1) W T1 S (V T

1 Z 1)

¼ ~ W T1 (S 1 S 01 V T1 Z 1) þ W T1 S 01 ~ V

T1 Z 1 þ d u 1 (63)

where S 1 ¼ S ( V T1 Z 1), S 01 ¼ diag {s 01, s 02, . . . , s 0l } with

^s

0

i ¼ s 0

(^v

T

i Z ) ¼

d[s (za )]

dza jza ¼v Ti Z

the residual term d u 1 is bounded by

jd u 1j kV 1 kF kZ 1 W T1 S 01kF þ kW 1 kkS 01 V T1 Z 1k þ jW 1 j1

(64)

and the weight estimation errors ~ W 1 ¼ W 1 W 1 , ~ V 1 ¼

V 1 V 1 .

Consider the following Lyapunov function candidate

V 1(r 1, ~ W 1, ~ V 1) ¼1

2d 11r 21 þ

1

2~ W 1

TG

1W 1

~ W 1 þ1

2tr{ ~ V 1

TG

1V 1

~ V 1}

(65)

The time derivative of (65) along (60) and (62) is given by

_V 1 ¼ r 1 b 11(_q 3)t 1 W T1 S 1(V T

1 Z 1) 11(Z 1)h i

þ ~ W 1 TG

1W 1

_~ W 1 þ tr{ ~ V 1 TG

1V 1

_~ V 1} (66)

As W 1 , V 1 are constant vectors, it is easy to obtain that

_~ W 1 ¼ _W 1, _~ V 1 ¼

_V 1 (67)

Substituting (67) into (66), we have

_V 1 ¼ r 1 b 11(_q 3)t 1 W T1 S 1(V T

1 Z 1) 11(Z 1)h i

þ ~ W 1 TG

1W 1

_W 1 þ tr{ ~ V 1

TG

1V 1

_V 1} (68)

Consider the following MNN-based control law and MNN weight adaption laws

t 1 ¼ k1r 1 r 1( W T1 S ( V T1 Z 1))2

b 11(jr 1 W T1 S ( V T1 Z 1)j þ d 1)

k1r 1b 11

(kZ 1 W T1 S 01k2F þ kS 01 V T1 Z 1k

2) (69)

_W 1 ¼ GW 1[(S 1 S 01 V T1 Z 1)r 1 þ s W 1

W 1] (70)

_V 1 ¼ GV 1[Z 1 W T1 S 01r 1 þ s V 1

V 1] (71)

where k1 . 0, d 1 . 0, GW 1 ¼ G TW 1 . 0, GV 1 ¼ G

TV 1 . 0,

s W 1 . 0, s V 1 . 0.

Substituting (69)–(71) in (68), we have

_V 1 ¼ k1b 11(_q 3)r 21 b 11(_q 3)

b 11

r 21 W T1 S ( V T1 Z 1) 2

jr 1 W T1 S ( V T1 Z 1)j þ d 1

b 11(_q 3)

b 11

k1r 21 kZ 1 W T1 S 01k2F þ kS 01 V T1 Z 1k

2

r 1W T1 S 1(V T1 Z 1) r 111(Z 1) r 1 ~ W T

1 (S 1 S 01 V T1 Z 1)

s W 1 ~ W

T

1 W 1 tr{ ~ V 1

TZ 1 W T1 S 01r 1} s V 1tr{ ~ V 1

TV 1}

(72)

Noting Assumption 3 and the fact that tr{ ~ V 1 T

Z 1 W T1 S 01r 1} ¼

r 1 W T1 S 01 ~ V 1 T

Z 1, (72) becomes

_V 1 k1b 11r 21 r 21 W T1 S ( V T1 Z 1) 2

jr 1 W T1 S ( V T1 Z 1)j þ d 1

k1r 21 kZ 1 W T1 S 01k2F þ kS 01 V T1 Z 1k2

þ jr 1jj11(Z 1)j

r 1W T1 S 1(V T

1 Z 1) r 1 ~ W T

1 (S 1 S 01 V T1 Z 1)

r 1 W T1 S 01 ~ V 1 T

Z 1 s W 1 ~ W

T

1 W 1 s V 1tr{ ~ V 1

TV 1} (73)

From (63) and (64), we know

r 1W T1 S 1(V T

1 Z 1) r 1 ~ W T

1 (S 1 S 01 V T1 Z 1) r 1 W T1 S 01 ~ V 1 T

Z 1

¼ r 1 W T1 S ( V T1 Z 1) r 1d u 1

jr 1 ^W

T

1 S ( ^V

T

1 Z 1)j þ jr 1jkV

1 kF kZ 1 ^W

T

1 ^S

0

1kF

þ jr 1jkW 1 kkS 01 V T1 Z 1k þ jr 1jjW 1 j1 (74)



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Substituting (74) into (73) leads to

_V 1 k1b 11r 21 r 21 W T1 S ( V T1 Z 1) 2

jr 1 W T1 S ( V T1 Z 1)j þd 1 þ jr 1 W T1 S ( V T1 Z 1)j


2

þ jr 1jj11(Z 1)j

þ jr 1jkV 1 kF kZ 1 W T1 S 01kF þ jr 1jkW 1 kkS 01 V T1 Z 1k

þ jr 1jjW 1 j1 s W 1 ~ W

T

1 W 1 s V 1tr{ ~ V 1

TV 1} (75)

According to Lemma 1,

r 21 W T1 S ( V T1 Z 1) 2

jr 1 W T1 S ( V T1 Z 1)j þd 1þ jr 1 W T1 S ( V T1 Z 1)j

¼jr 1 W T1 S ( V T1 Z 1)jd 1

jr 1 W T1 S ( V T1 Z 1)j þd 1 d 1 (76)


jr 1jj11(Z 1)j r 21

2c 11

þc 11 1

21

2 (77)

jr 1jkV 1 kF kZ 1 W T1 S 01kF k1r 21 kZ 1 W T1 S 01k2F þ

1

4k1

kV 1 k2F

(78)

jr 1jkW 1 kkS 01 V T1 Z 1k k1r 21 kS 01 V T1 Z 1k2

þ1

4k1

kW 1 k2 (79)

jr 1jjW 1 j1 r 21

2c 12

þc 12jW 1 j

21

2 (80)

s W 1 ~ W

T

1 W 1

s W 1

2 k ~ W 1k

2þs W 1

2 kW 1k

2 (81)

s V 1tr{ ~ V 1 T

V 1} s V 1

2 k ~ V 1k

2F þ

s V 1

2 kV 1 k

2F (82)

Substituting (76)–(82) into (85), we have

_V 1 k1b 11 1

2c 11

1

2c 12

r 21

s W 1

2 k ~ W 1k

2

s V 1

2 k ~ V 1k

2F þd 1 þ

s W 1

2 þ

1

4k1

kW 1 k

2

þs V 1

2 þ

1

4k1

kV 1 k

2F þ

c 11

2 1

21 þ

c 12jW 1 j21

2

l10V 1 þm10 (83)

where l10¼

min{(2k1b 11

1=c 11

1=c 12)=d 11, s W 1=lmax(G1

W 1),s V 1=lmax(G1V 1)},m10 ¼ d 1 þ(s W 1=2 þ 1=4k1)kW 1 k

2þ

(s V 1=2 þ 1=4k1)kV 1 k2F þ (c 11=2)1

21 þ (c 12jW 1 j

21=2).

3.2.2 q2 subsystem: Similar to Section 3.1.2, (10)becomes

d 22(q 3)d 33 d 223

d 33

_r 2 þ c 22(q 3, _q 3)r 2 ¼ b 22(_q 3)t 2 f S 2,1 (84)

where the unknown function

f S 2,1 ¼d 22(q 3)d 33 d 223

d 33

€q 2r þ c 22(q 3, _q 3)_q 2r

þ c 23(q 3, _q 2)_q 3 þ D2(q , _q ) þd 23

d 33

(b 31(_q 3)t 1

c 32(q 3, _q 2)_q 2 f 3(_q 3) g 3 D3(q , _q ))

is approximated by MNN to arbitrary any accuracy as

f S

2,1 ¼ W T2 S 2(V T

2 Z 2) þ 12(Z 2)

where the input vector Z 2 ¼ [t 1, q 1, _q 1, q 2, _q 2, q 3, _q 3,q 2d , _q 2d , €q 2d , 1] T [ VZ 2

, R 11, 12(Z 2) is the approximationerror satisfying j12(Z 2)j 12, where 12 is a positive constant;W 2 and V 2 are ideal constant weights satisfying kW 2 k

w2m, kV 2 kF v 2m, which are positive constants. By using

W T2 S 2( V T2 Z 2) to approximate W T2 S 2(V T

2 Z 2), the error between the actual and the ideal MNN can be expressed as

W T2 S ( V T2 Z 2) W T2 S (V T

2 Z 2)

¼ ~ W T

2 (S 2 S 02 V T2 Z 2) þ W T2 S 02 ~ V T

2 Z 2 þ d u 2 (85)

where S 2 ¼ S ( V T2 Z 2), S 02 ¼ diag {s 01, s 02, . . . , s 0l } with

s 0i ¼ s 0(v Ti Z 2) ¼d [s (za )]

dza

jza ¼v Ti Z 2

the residual term d u 2 is bounded by

jd u 2j kV 2 kF kZ 2 W T2 S 02kF þ kW 2 kkS 02 V T2 Z 2k þ jW 2 j1

(86)

and the weight estimation errors ~ W 2 ¼ W 2 W 2 , ~ V 2 ¼

V 2 V 2 .

To analyse the closed-loop stability for the q 2-subsystem,consider the following Lyapunov function candidate

V 2(r 2, ~ W 2, ~ V 2) ¼1

2

d 22(q 3)d 33 d 223

d 33

r 22

þ1

2~ W

T

2 G1W 2

~ W 2 þ1

2tr{ ~ V

T

2 G1V 2

~ V 2} (87)

Lemma 5: The function V 2 (87) is positive definite anddecrescent, in the sense that there exist two time-invariant positive definite functions V 2(r 2, ~ W 2, ~ V 2) and V 2(r 2,

~ W 2, ~ V

2), such that

V 2(r 2, ~ W 2, ~ V 2) V 2 V 2(r 2, ~ W 2, ~ V 2)



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Proof: The proof can be referred to that of Lemma 3 andomitted here for conciseness. A

The time derivative of (87) is given as

_V 2 ¼

1

2_d 22(q 3)r

22 þ

d 22(q 3)d 33 d 223

d 33r 2 _r 2

þ ~ W T2 G

12

_~ W 2 þ tr{ ~ V T2 G

1V 2

_~ V 2} (88)

According to Assumption 2, (88) becomes

_V 2 ¼ r 2d 22(q 3)d 33 d 223

d 33

_r 2 þ c 22(q 3, _q 3)r 2

" #

þ ~ W T

2G12

_~ W 2 þ tr{ ~ V T

2 G1V 2

_~ V 2} (89)

As W 2 , V 2 are constant vectors, it is easy to obtain that

_~ W 2 ¼ _W 2, _~ V 2 ¼

_V 2 (90)

Substituting (84), (85) and (90) into (94), we have

_V 2 ¼ r 2 b 22(_q 3)t 2 W T2 S 2(V T

2 Z 2) 12(Z 2)h i

þ ~ W T

2 G1W 2

_W 2 þ tr{ ~ V

T

2 G1V 2

_V 2} (91)

Consider the following MNN-based control law and MNN weight adaption laws

t 2 ¼ k2r 2 þr 2 W T2 S ( V T2 Z 2) 2

b 22 jr 2 W T2 S ( V T2 Z 2)j þ d 2

þk2r 2b 22

(kZ 2 W T2 S 02k2F þ kS 02 V T2 Z 2k

2) (92)

_W 2 ¼ GW 2[(S 2 S 02 V T2 Z 2)r 2 þ s W 2

W 2] (93)

_V 2 ¼ GV 2[Z 2 W T2 S 02r 2 þ s V 2

V 2] (94)

where k2 . 0, d 2 . 0, GW 2 ¼ G TW 2 . 0, GV 2 ¼ G

TV 2 . 0,

s W 2 . 0, s V 2 . 0.


_V 2 ¼ k2b 22(_q 3)r 22 þb 22(_q 3)

b 22

r 22 W T2 S ( V T2 Z 2) 2

jr 2 W T2 S ( V T2 Z 2)j þ d 2

þb 22(_q 3)

b 22


2

r 2W T2 S 2(V T

2 Z 2) r 212(Z 2) r 2 ~ W T

2 (S 2 S 02 V T2 Z 2)

s W 2 ~ W T2 W 2 tr{ ~ V 2 TZ 2 W T2 S 02r 2} s V 2tr{ ~ V 2 TV 2}

(95)

Noting Assumption 3 and the fact that tr { ~ V 2 T

Z 2 W T2 S 02r 2} ¼

r 2 W T2 S 02 ~ V 2 T

Z 2, (95) becomes

_V 2 k2b 22r 21 r 22 W T2 S ( V T2 Z 2) 2

jr 2 ^

W T2 S (

^

V T2 Z 2)j þ d 2

k2r 22 kZ 2 W T2 S 02k

2F þ kS 02 V T2 Z 2k

2

þ jr 2jj12(Z 2)j

r 2W T2 S 2(V T

2 Z 2) r 2 ~ W T

2 (S 2 S 02 V T2 Z 2)

r 2 W T2 S 02 ~ V 2 T

Z 2 s W 2 ~ W

T

2 W 2 s V 2tr{ ~ V 2

TV 2} (96)

From (85) and (86), we know

r 2W T2 S 2(V T

2 Z 2) r 2 ~ W T

2 (S 2 S 02 V T2 Z 2) r 2 W T2 S 02 ~ V 2 T

Z 2

¼ r 2 W T2 S ( V T2 Z 2) r 2d u 2

jr 2 W T2 S ( V T2 Z 2)j þ jr 2jkV 2 kF kZ 2 W T2 S 02kF

þ jr 2jkW 2 kkS 02 V T2 Z 2k þ jr 2jjW 2 j1 (97)

Substituting (97) to (96) leads to

_V 2 k2b 22r 22 r 22 W T2 S ( V T2 Z 2) 2

jr 2 W T2 S ( V T2 Z 2)j þd 2

þ jr 2 W T2 S ( V T2 Z 2)j

k2r 22 kZ 2

^W

T2

^S

0

2k2F þ k

^S

0

2 ^V

T2 Z 2k

2 þ jr 2jj12(Z 2)j þ jr 2jkV 2 kF kZ 2 W T2 S 02kF

þ jr 2jkW 2 kkS 02 V T2 Z 2k þ jr 2jjW 2 j1

s W 2 ~ W

T

2 W 2 s V 2tr{ ~ V 2

TV 2} (98)

According to Lemma 1

r 22 W T2 S ( V T2 Z 2) 2

jr 2 ^W

T2 S (

^V

T2 Z 2)j þd 2

þ jr 2 W T2 S ( V T2 Z 2)j

¼jr 2 W T2 S ( V T2 Z 2)jd 2

jr 2 W T2 S ( V T2 Z 2)j þd 2 d 2 (99)


jr 2jj12(Z 2)j r 22

2c 21

þc 21 1

22

2 (100)

jr 2jkV 2 kF

kZ 2 W T2 S 02kF

k2r 22 kZ 2 W T2 S 02k2

F

þ1

4k2

kV 2 k2

F

(101)



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jr 2jkW 2 kkS 02 V T2 Z 2k k2r 22 kS 02 V T2 Z 2k2

þ1

4k2

kW 2 k2

(102)

jr 2jjW 2 j1 r 22

2c 22

þc 22jW 2 j

21

2

(103)

s W 2 ~ W

T

2 W 2

s W 2

2 k ~ W 2k

2þs W 2

2 kW 1k

2 (104)

s V 2tr{ ~ V 2 T

V 2} s V 2

2 k ~ V 2k

2F þ

s V 2

2 kV 2 k

2F (105)


_V 2 k2b 22 1

2c 21

1

2c 22

r 22

s W 2

2 k ~ W 2k

2

s V 2

2

k ~ V 2k2F þd 2 þ

s W 2

2

þ1

4k2 kW 2 k

2

þs V 2

2 þ

1

4k2

kV 2 k

2F þ

c 21

2 1

22 þ

c 22jW 2 j21

2

l20V 2 þm20 (106)

where l20 ¼ min{(2k2b 22 1=c 21 1=c 22)jd 33j=( d 22jd 33jþ

d 223), s W 2=lmax(G1W 2),s V 2=lmax(G1

V 2)}, m20 ¼ d 2 þ (s W 2=

2 þ 1=4k2)kW 2 k2

þ(s V 2=2 þ 1=4k21)kV 2 k2F þ(c 21=2)1

22 þ c 22

jW 2 j21=2.

3.2.3 q3 subsystem: Finally, for the system (9)– (11)under control laws (69), (92), we can obtain the similar internal dynamics to Section 3.1.3.

The main result in this section is summarised in thefollowing Theorem.

Theorem 2: Consider the closed-loop system consisting of the subsystems (9)–(11), the control laws (69), (92), andadaptation laws (70), (71), (93) and (94). Under

Assumptions 1– 8, the overall closed-loop neural controlsystem is SGUUB in the sense that all of the signals in the

closed-loop system are bounded, and the tracking errorsand weights converge to the following regions

je 1j je 1(0)j þ1

l1


d 11

s , k W 1k


lmin(G11 )

s þ w1m

kV 1kF


lmin(G11 )

s þ v 1m

je 2j je 2(0)j þ1

l2


jd 22jd 33j d 223j

s

k W 2k


lmin(G12 )

s þ w2m, kV 2k


lmin(G12 )

s þ v 2m

with

mi ¼mi 0

li 0

þ V i (0)

mi 0 ¼ d i þs Wi

2

þ1

4ki kW i k

2þ

s Vi

2

þ1

4ki 1 kV i k

2F

þc i 12

12i þ

c i 2jW i j21

2 , i ¼ 1, 2

l10 ¼ min{(2k1b 11 1=c 11 1=c 12)=d 11

s W 1=lmax(G1W 1), s V 1=lmax(G1

V 1)}

l20 ¼ min{(2k2b 22 1=c 21 1=c 22)jd 33j=( d 22jd 33j þ d 223)

s W 2=lmax(G1W 2), s V 2=lmax(G1

V 2)}

where e i (0) and V i (0) are initial values of e i (t ) and V i (t ),respectively.

Proof: The proof of Theorem 2 follows the same approach as Theorem 1, and is omitted here for conciseness. A

4 Simulation study

To illustrate the proposed adaptive neural control, we consider the VARIO helicopter mounted on a platform [6], with thedynamic model as (2) and the following parameters d 11 ¼ 7:5,d 22(q 3) ¼ 0:4305þ0:0003cos2(4:143q 3), d 23 ¼ 0:108, d 33 ¼

0:4993, c 22(q 3, _q 3) ¼ 0:0006214 sin(8:286q 3)_q 3, c 23(q 3,_q 2) ¼

c 32(q 3,_q 2) ¼ 0:0006214 sin (8:286q 3)_q 2, f 1(_q 3) ¼ 0:6004_q 3,

f 3(_q 3) ¼ 0:0001206_q 23, g 1 ¼ 77:259, g 3 ¼ 2:642, b 11(_q 3) ¼

3:411_q 23, b 22(_q 3) ¼ 0:1525 _q 23, b 31(_q 3) ¼ 12:01_q 3 þ 105, andall quantities are expressed in SI units. The control objective is totrack the uniformly bounded desired trajectories given in [6] asfollows

q 1d ¼

0:2 0 t 50s

0:3[e(t 50)2=350

1] 0:2 50 , t 130s

0:1cos[(t 130)=10] 0:6 130 , t 20p þ 130

0:5 t 20p þ 130

8>>><>>>:

q 2d ¼

0 t , 50s

1 e(t 50)2=350 50 t , 120s

e(t 120)2=350 120 t , 180

1 þ e(t 180)2=350 t 180

8>>><>>>:

4.1 Internal dynamics stability analysis

In this section, we analyse the stability of the internaldynamics. For conciseness, we consider the RBFNN-basedcontrol case only, which can be easily extended to MNN-based control case without any difficulties. For the

RBFNN based control case, we substitute (15), (20), (30)and (38) into the q 3-subsystem (11). According to the

definition of the zero-dynamics [26], we set r 1, r 2, ~ W T

1 ,



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~ W T

2 , 11(Z 1) and 12(Z 2) to zero, and that the desiredtrajectories and initial data can be chosen in such a way that terms including _q 22, €q 1d , €q 2d can be neglected [6], wehave

€q 3 ¼ 1d 33

b 31(_q 3)b 11(_q 3)

( f 1(_q 3) þ g 1) f 3(_q 3) g 3

(107)

Noticing that the zero-dynamics in (107) is nonlinear, it isdifficult to conduct the stability analysis of the nonlinear zero-dynamics directly in this example. Instead, we show its local stability within an interested operating region after linearising the nonlinear zero-dynamics around anequilibrium operating point of interest. Although thestability of the linearised system cannot exactly guaranteethe stability of the nonlinear zero-dynamics, the localstability is at least ensured within the operating region of

interest. To realise this, we need to solve for theequilibrium points first.

Substituting the term values given in the beginning of Section 4 into (107) and analysing the values of the mainrotor angular velocity from which the main rotor angular acceleration is zero, we have

4:1137 104_q 43 þ 1:8011_q 23 60968_q 3 7725900 ¼ 0

(108)

Its solutions are _q 3 ¼ 124:63, 219:5+ 468:16i and563.64 rad/s. Only the first value _q 3 ¼ 124:63 has a physical meaning for the system (see Fig. 1 for the rotationsense of the main rotor). If we linearise equation (107)around the equilibrium point _q 3 ¼ 124:63, we can obtainan eigenvalue 22.44. Therefore according to [27], all initial

values of _q 3 sufficiently near the equilibrium point _q 3 ¼ 124:63 can converge to 2124.63, indicating that thezero-dynamics of the helicopter system in (2) has a locally stable behaviour around the equilibrium point. Thelimitation of this analysis is that if the initial conditions arefar from the operating region of interest, the stability of the

zero-dynamics is not guaranteed.

The simulation result in Fig. 2 also shows that the zero-dynamics using RBFNN-based control are locally stablearound the equilibrium point _q 3 ¼ 124:63. From Fig. 2,

we can observe that the main rotor angular velocity _q 3converges to the nominal value 2124.63 rad/s for different initial conditions ranging from 240 to 2150 rad/s, whichincludes the typical operating values more than sufficiently.

These results are expected from the previous stability analysis, and also consistent with the results in [6]. Inparticular, we also notice that the further the initial

condition starts from the nominal value2

124.63 rad/s, thelonger the settling time takes, and the more seriously thetransient oscillations become.

4.2 Performance comparison resultsbetween approximation-based control and model-based control

In this subsection, we will compare the altitude and yaw angle tracking performance using RBFNN-based control,

MNN-based control and model-based control adoptedin [6]. If all the parameters and functions in (2) areknown exactly, and the unmodelled uncertaintiesD() ¼ 0, the perfect tracking performance can beachieved using model-based control, which has beenshown in the work [6]. However, in the practice, therealways exist some model uncertainties, which may becaused by unmodelled dynamics or aerodynamicaldisturbances from the environment. To this end, weassume D()= 0, in particular, D() ¼ [2:0, 0,0:0001206_q 23 þ 0:142] T.

The control parameters for the RBFNN control laws (20)(38) and adaptation laws (21) (39) are chosen as follows:k1 ¼ 0:000085, L1 ¼ 0:2, k2 ¼ 0:0002, L2 ¼ 1:0,G1 ¼ 0:001 I , G2 ¼ 0:0001 I , s 1 ¼ 0:001, s 2 ¼ 0:001.NNs W T1 S 1(Z 1) contain 38 nodes (i.e. l 1 ¼ 2187),

with centres ml (l ¼ 1, . . . , l 1) evenly spaced in[1:0, 1:0] [0:1, 0:1] [10:0, 10:0] [40000:0,0:0][1:0, 1:0] [150:0,40:0] [0:1, 0:1][0:01,0:01], and widths h l ¼ 1:0(l ¼ 1, . . . , l 1). NNs W T2 S 2(Z 2)

contain 310 nodes (i.e. l 2 ¼ 59049), with centres ml (l ¼

1, . . . , l 2) evenly spaced in [0:005, 0:005] [1:0, 1:0]

[0:1, 0:1] [10:0, 10:0] [ 40000:0, 0:0] [1:0,

1:0] [150:0, 40:0] [10:0, 10:0] [ 1:0, 1:0][0:01, 0:01], and widths h l ¼ 1:0(l ¼ 1, . . . , l 2). Theinitial conditions are q 1(0) ¼ 0:1 m, _q 1(0) ¼ 0:0 m/s,q 2(0) ¼ p rad, _q 2(0) ¼ 0:0 rad/s, q 3(0) ¼ p rad, _q 3(0) ¼

120:0 rad/s, t 1 ¼ 0:0 m, t 2 ¼ 0:0 m, W 1(0) ¼ 0:0,W 2(0) ¼ 0:0.

Figure 2 Local stability of the zero-dynamics around theequilibrium point q

3

¼ 2124.63 using RBFNN-based

control



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For the MNN control laws (69) and (92) and adaptationlaws (70), (71), (93) and (94), the design parameters arechosen as k1 ¼ 0:00016, L1 ¼ 1:2, k2 ¼ 0:0002, L2 ¼ 1:0,GW 1 ¼ 0:0002 I , GV 1 ¼ 0:03 I , d W 1 ¼ 0:0, s V 1 ¼ 0:0,GW 2 ¼ 0:0001 I , GV 2 ¼ 0:01 I , s W 2 ¼ 0:0, s V 2 ¼ 0:0.NNs W T1 S 1( V T1 z1) contain five nodes and NNs

W T2 S 2( V T2 z2) contains 15 nodes. The initial conditions areq 1(0) ¼ 0:1 m, _q 1(0) ¼ 0 m/s, q 2(0) ¼ p rad, _q 2(0) ¼

0:0 rad/s, q 3(0) ¼ p rad, _q 3(0) ¼ 120:0 rad/s, t 1 ¼

0:0 m, t 2 ¼ 0:0 m, W 1(0) ¼ 0:0, V 1(0) ¼ 0:0, W 2(0) ¼ 0:0,V 2(0) ¼ 0:0.

From Figs. 3 and 4, we can observe that due to theexistence of model uncertainties, both the altitudetracking and yaw angle tracking using model-basedcontrol have some offsets to the desired trajectories for the whole period. It means that model-based controldepends on the accuracy of model heavily and cannot deal with the uncertainties well. For the tracking performance using the RBFNN-based control andMNN-based control, although there are also someoscillations at the beginning period, the tracking errorscan converge to a very small neighbourhood of desired

Figure 3 Altitude tracking performance in the presence of

model uncertainties using three methods: RBFNN-based

control, MNN-based control and model-based control,

respectively a Zoom-outb Zoom-in

Figure 5 Control inputs for altitude and yaw angle tracking

in the presence of model uncertainties using three methods:

RBFNN-based control, MNN-based control and model-

based control, respectively

Figure 4 Yaw angle tracking performance in the presence

of model uncertainties using three methods: RBFNN-based

control, MNN-based control and model-based control,

respectively a Zoom-outb Zoom-in

Figure 6 Norm of neural weights using RBFNN-based

control and MNN-based control



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trajectories in a short time about 20 s. This is because themodel uncertainties can be learnt by RBFNN and MNNduring the beginning 20 s. After that period, theuncertainties can be compensated for, and thus, therobustness to uncertainties is improved and the goodtracking performance is achieved. In addition, Figs. 5

and 6 indicate the boundedness of the control actionsand neural weights for all control methods.

5 Conclusion

In this paper, NN approximation-based control has beeninvestigated for the helicopter altitude and yaw angletracking in the presence of model uncertainties. Compared

with the model-based control, which is sensitive to theaccuracy of the model representation, NN approximationbased control is tolerant of model uncertainties, and can be

viewed as a key advantage over model-based control of helicopters, for which accurate modelling of helicopter dynamics is difficult, time-consuming and costly.Simulation results demonstrated that the helicopter is ableto track altitude and yaw angle reference signalssatisfactorily, with all other closed-loop signals bounded.

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