RBFNN-Based Control

5.3 Control Design

5.3.1 RBFNN-Based Control

In this section, we will investigate the RBFNN based control design by Lyapunov synthesis to achieve the control objective. Regarding to the obtained three subsys- tems (5.4)–(5.6), our control design consists of three steps: First, we will design control1 based on the q1-subsystem (5.4); Second, design2 based on the q2- subsystem (5.5) and 1; finally, analyze the stability of the internal dynamics of q3-subsystem (5.6).

q1-subsystem

SinceqP1D Pq1r Cr1,qR1D Rq1r C Pr1, (5.4) becomes

d11rP1Db11.qP3/1fS1;1 (5.8) where

fS1;1Dd11qR1rCf1.qP3/Cg1C1.q;q/P (5.9) is an unknown continuous function, which is approximated by RBFNN to arbitrarily any accuracy as

fS1;1DW₁^TS1.Z1/C"1.Z1/ (5.10) where the input vectorZ1 D Œq1; qP1; q2; qP2; q3; qP3; qP1d; qR1d^T 2 ˝Z1 R⁸;

"1.Z1/is the approximation error satisfyingj"1.Z1/j N"1, where"N1 is a positive constant;W₁are ideal constant weights satisfyingkW₁k w1m, where w1mis a positive constant; andS1.Z1/are the basis functions. By usingWO1 to approximate W₁, the error between the actual and the ideal RBFNNs can be expressed as

WO1

TS1.Z1/W₁^TS1.Z1/D QW1

TS1.Z1/ (5.11) whereWQ₁D OW₁W₁.

Consider the following Lyapunov function candidate V1D 1

2d11r₁²C1 2WQ1

T 1

1 WQ1 (5.12)

The time derivative of (5.12) along (5.8) and (5.10) is given by VP₁Dd₁₁r₁rP₁C QW₁^T ₁¹WPQ₁

Dr1

b11.qP3/1W₁^TS1.Z1/"1.Z1/ C QW1

T 1

1 WPQ1 (5.13)

98 5 Altitude and Yaw Control of Helicopters with Uncertain Dynamics

AsW₁is a constant vector, we know thatWPQ₁D POW₁. Therefore, (5.13) becomes VP1Dr1

b11.qP3/1W₁^TS1.Z1/"1.Z1/ C QW1

T 1

1 WPO1 (5.14) Consider the following RBFNN-based control law and RBFNN weight adaptation law:

1D k1r1 r1

WO1

TS1.Z1/ 2

b₁₁

jr₁WO₁^TS₁.Z₁/j Cı₁

(5.15)

WPO₁D 1

h

S₁.Z₁/r₁C₁WO₁ i

(5.16) wherek1> 0,ı1> 0, 1D ₁^T> 0, and1> 0.

Remark 5.5. The above-modification adaptation law (5.16) can be replaced by e-modification adaptation law likeWPO1 D 1

h

S1.Z1/r1C1jr1j OW1

i

easily. The control design based on -modification adaptation law in this chapter can be extended to the case based one-modification adaptation law without any difficulty.

Substituting (5.15) and (5.16) into (5.14), we have

VP₁ D k₁b₁₁.qP₃/r₁²b11.qP3/ b₁₁

r₁²

WO₁^TS₁.Z₁/ 2

jr1WO1

TS1.Z1/j Cı1

r₁W₁^TS₁.Z₁/r₁"₁.Z₁/ r1WQ1

TS1.Z1/1WQ1

TWO1 (5.17)

According to Assumption5.3and (5.11), we can rewrite (5.17) as

VP1 k1b₁₁r₁² r₁² WO1

TS1.Z1/ 2

jr1WO1

TS1.Z1/j Cı1

r1WO1

TS1.Z1/r1"1.Z1/

1WQ1 TWO1

k₁b₁₁r₁² r₁² WO1

TS1.Z1/ 2

jr₁WO₁^TS₁.Z₁/j Cı₁ C jr₁WO₁^TS₁.Z₁/j C jr₁jj"₁.Z₁/j

₁WQ₁^TWO₁ (5.18)

Noting that

r₁²

WO₁^TS₁.Z₁/ 2

jr1WO1

TS1.Z1/j Cı1

C jr1WO1

TS1.Z1/j D jr1WO1

TS1.Z1/jı1

jr1WO1

TS1.Z1/j Cı1

(5.19)

According to Lemma5.4, we can obtain from (5.19) that

r₁² WO1

TS1.Z1/ 2

jr1WO1

TS1.Z1/j Cı1

C jr1WO1

TS1.Z1/j ı1 (5.20)

By completion of squares and using Young’s inequality, the following inequali- ties hold:

₁WQ₁^TWO₁ 1

2 k QW₁k²C 1

2kW₁k² (5.21)

jr1jj"1.Z1/j r₁²

2c₁ Cc1"²₁.Z1/ 2 r₁²

2c₁ C c1"N²₁

2 (5.22)

wherec₁is a positive constant. Substituting the above inequalities (5.20)–(5.22) into (5.18) leads to

VP1

k1b₁₁ 1 2c₁

r₁²₁

2 k QW1k²Cı1C c₁ 2"N²₁C ₁

2w²_1m

10V1C10 (5.23)

where10 Dmin n

.2k1b₁₁1=c1/=d11; 1=max. ₁¹/ o

,10Dı1C^c₂¹"N²₁C₂¹w²_1m. q₂-subsystem

Similar to Sect.5.3.1, sinceqP₂D Pq_2rCr₂,qR₂D Rq_2rC Pr₂, (5.5) becomes d₂₂.q₃/d₃₃d₂₃²

d₃₃ rP2Cc22.q3;qP3/r2 Db22.qP3/2fS 2;1 (5.24) where

fS 2;1D d₂₂.q₃/d₃₃d₂₃²

d33 qR2rCc22.q3;qP3/qP2rCc23.q3;qP2/qP3C2.q;q/P Cd23

d₃₃.b31.qP3/1c32.q3;qP2/Pq2f3.Pq3/g33.q;q//P

is an unknown function, which is approximated by RBFNN to arbitrarily any accuracy as

fS 2;1DW₂^TS2.Z2/C"2.Z2/ (5.25) where the input vector Z2 D Œ1; q1; qP1; q2; qP2; q3; qP3; q2d; qP2d; qR2d^T 2

˝Z₂ R¹⁰,"2.Z2/is the approximation error satisfyingj"2.Z2/j N"2, where"N2

is an unknown positive constant;W₂are unknown ideal constant weights satisfying kW₂k w2m, where w2m is an unknown positive constant; andS2.Z2/are the

100 5 Altitude and Yaw Control of Helicopters with Uncertain Dynamics

basis functions. By usingWO2to approximateW₂, the error between the actual and the ideal RBFNNs can be expressed as

WO₂^TS2.Z2/W₂^TS2.Z2/D QW₂^TS2.Z2/ (5.26) whereWQ2D OW2W₂.

To analyze the closed loop stability for theq2-subsystem, let V₂D 1

2

d22.q3/d33d₂₃² d33

r₂²C 1

2WQ₂^T ₂¹WQ₂ (5.27) Lemma 5.6. The function V2 (5.27) is positive definite and decrescent, in the sense that there exist two time-invariant positive definite functionsV₂.r2;WQ2/and VN2.r2;WQ2/, such that

V₂.r2;WQ2/V2 NV2.r2;WQ2/

Proof. Noting that the particular choice ofV2 in (5.27), a function ofr2;WQ2 and d22.q3/, is to establish the stability forr2andWQ2only, therefore, we regardd22.q3/ as a function of time. From Assumptions5.1and5.4, we know that

0 <

ˇˇˇd₂₂jd₃₃j d₂₃²ˇˇˇ

jd33j <ˇˇˇd22.q3/d33d₂₃² d33

ˇˇˇ dN22jd33j Cd₂₃²

jd33j (5.28) Therefore, there also exist time-invariant positive definite functionsV₂.r2;WQ2/ andVN2.r2;WQ2/, such thatV₂.r2;WQ2/ V2 NV2.r2;WQ2/, which implies thatV2is also positive definite and decrescent, according to [94]. This completes the proof.ut The time derivative of (5.27) is given as

VP2D 1

2dP22.q3/r₂²C d₂₂.q₃/d₃₃d₂₃²

d₃₃ r2rP2C QW₂^T ₂¹WPQ2 (5.29) According to Assumption5.2, (5.29) becomes

VP2 Dr2

d₂₂.q₃/d₃₃d₂₃² d33

P

r2Cc22.q3;qP3/r2 C QW₂^T ₂¹WPQ2 (5.30) AsW₂is a constant vector, it is easy to obtain that

WPQ2D POW2 (5.31)

Substituting (5.24), (5.25) and (5.31) into (5.30), we have VP₂Dr₂

b₂₂.qP₃/₂W₂^TS₂.Z₂/"₂.Z₂/

C QW₂^T ₂¹WPO₂ (5.32) Consider the following RBFNN-based control law and RBFNN weight adaption law:

2 Dk2r2C r₂

WO₂^TS₂.Z₂/ 2

b₂₂ jr2WO2

TS2.Z2/j Cı2

(5.33)

WPO2 D 2

h

S2.Z2/r2C2WO2

i

(5.34) wherek₂> 0,ı₂> 0, ₂D ₂^T > 0and₂> 0. Substituting (5.33) and (5.34) into (5.32), we have

VP₂Dk₂b₂₂.qP₃/r₁²C b22.qP3/ b₂₂

r₂² WO2

TS2.Z2/ 2

jr₂WO₂^TS₂.Z₂/j Cı₂ r₂W₂^TS₂.Z₂/r₂"₂.Z₂/

r₂WQ₂^TS₂.Z₂/₂WQ₂^TWO₂ (5.35)

According to Assumption5.3and (5.26), we can rewrite (5.35) as

VP2 k2b₂₂r₂² r₂²

WO₂^TS₂.Z₂/ 2

jr2WO2

TS2.Z2/j Cı2

r2WO2

TS2.Z2/r2"2.Z2/

2WQ2 TWO2

k2b₂₂r₂² r₂² WO2

TS2.Z2/ 2

jr2WO2

TS2.Z2/j Cı2

C jr2WO2

TS2.Z2/j C jr2jj"2.Z2/j

2WQ2

TWO2 (5.36)

Similar to (5.20), we have

r₂² WO2

TS2.Z2/ 2

jr2WO2

TS2.Z2/j Cı2

C jr2WO2

TS2.Z2/j ı2 (5.37)

102 5 Altitude and Yaw Control of Helicopters with Uncertain Dynamics By completion of squares and using Young’s inequality, the following inequali- ties hold:

₂WQ₂^TWO₂ 2

2 k QW₂k²C 2

2kW₂k² (5.38)

jr₂jj"₂.Z₂/j r₂²

2c2 Cc2"²₂.Z2/ 2 r₂²

2c2 C c2"N²₂

2 (5.39)

wherec2is a positive constant. Substituting the above inequalities (5.37)–(5.39) into (5.36) leads to

VP₂

k₂b₂₂ 1 2c2

r₂²2

2 k QW₂k²Cı₂Cc2

2"N²₂C2

2w²_2m

20V2C20 (5.40)

where20 D min n

.2k2b₂₂1=c2/jd33j=.dN22jd33j Cd₂₃²/; 2=max. ₂¹/ o

,20 D ı2C ^c₂²"N²₂C ₂²w²_2m.

q3-subsystem

Finally, using the designed control laws (5.15) and (5.33), theq3-subsystem (5.6) can be rewritten as

P

D .; ;u/ (5.41)

whereDŒq3;qP3^T, DŒq1; q2;qP1;qP2^T, uDŒ1; 2^T. Then, the zero dynamics can be addressed as [35]

P

D .0; ;u.0; // (5.42)

where uDŒ₁; ₂^T.

Assumption 5.6. [35] System (5.4)–(5.6) is hyperbolically minimum-phase, i.e., zero dynamics (5.42) is exponentially stable. In addition, assume that the control input u is designed as a function of the states (, ) and the reference signal satisfying Assumption5.5, and the functionf .; ;u/is Lipschitz in, i.e., there exist constantsLandLf forf .; ;u/such that

kf .; ;u/f .0; ;u/k Lkk CLf (5.43) where uDu.0; /.

Under Assumption5.6, by the Converse Theorem of Lyapunov [52], there exists a Lyapunov functionV0./which satisfies

akk² V0./bkk² (5.44)

@V₀

@f .0; ;u/ akk² (5.45)

k@V0

@k _bkk (5.46)

where_a,_b,_aand_b are positive constants.

Lemma 5.7. [35] For the internal dynamics P D f .; ;u/ of the system, if Assumption5.6 is satisfied, and the states are bounded by a positive constant kkmax, i.e.,kk kkmax, then there exist positive constantsLandT₀, such that

k.t /k L; 8t > T₀ (5.47) Proof. According to Assumption 5.6, there exists a Lyapunov function V0./.

DifferentiatingV0./along (5.4)–(5.6) yields VP0./D @V0

@f .; ;u/

D @V0

@f .0; ;u/C @V0

@

f .; ;u/f .0; ;u/

(5.48) Noting (5.43)–(5.46), (5.48) can be written as

VP0./ akk²Cbkk.Lkk CLf/

akk²Cbkk.LkkmaxCLf/ Therefore,VP₀./0, whenever

kk b

a

.LkkmaxCLf/

By letting L D ^b_a.LkkmaxC Lf/, we conclude that there exists a positive

constantT0, such that (5.47) holds. ut

The following Theorem shows the stability and control performance of the closed loop system.

Theorem 5.8. Consider the closed-loop system consisting of the subsystems (5.4)–(5.6), the control laws (5.15), (5.33) and adaptation laws (5.16), (5.34).

Under Assumptions5.1–5.6, the overall closed-loop neural control system is Semi- Globally Uniformly Ultimately Bounded (SGUUB) in the sense that all of the signals in the closed-loop system are bounded, and the tracking errors and neural weights converge to the following regions,

104 5 Altitude and Yaw Control of Helicopters with Uncertain Dynamics

je₁j je₁.0/j C 1 1

s 21

d11

k OW1k

s 2₁

_min. ₁¹/Cw1m

je₂j je₂.0/j C 1 2

vu

ut 2jd33j2

ˇˇˇd₂₂jd₃₃j d₂₃²ˇˇˇ k OW2k

s 22

min. ₂¹/Cw2m (5.49)

with

i D _{i 0}

_{i 0} CVi.0/; i 0DıiC 1 2"N²_i C _i

2w²_{i m}; iD1; 2 ₁₀Dmin

n

.2k₁b₁₁1=c₁/=d₁₁; ₁=_max. ₁¹/ o 20Dmin

n

.2k21=c2/jd33j=.dN22jd33j Cd₂₃²/; 2=max. ₂¹/ o

whereei.0/andVi.0/are initial values ofei.t /andVi.t /, respectively.

Proof. Based on the previous analysis, the proof proceeds by studying each subsys- tem in order. First, the closed loop stability analysis of theq₁-subsystem (5.4) with control₁(5.15) and adaptation law (5.16) is made by use of Lyapunov synthesis.

Second, the similar closed loop stability will be achieved on theq₂-subsystem (5.5) with₂ (5.33) and adaptation law (5.34). Finally, the stability analysis of internal dynamics of theq₃-subsystem (5.6) is made based on the stability of the previous two subsystems.

q₁- subsystem:

Solving the inequality (5.23), we have0 V₁.t /₁with₁ D ₁₀¹⁰ CV₁.0/.

Then, from the definition ofV₁.t /(5.12), we can obtain jr₁j

s 21

d11

; k QW₁k s

21

min. ₁¹/ (5.50)

SinceeP1D 1e1Cr1, solving this equation results in e1De^1te1.0/C

Z _t

0

e^{1.t /}r1d (5.51)

According to (5.50) and (5.51), we have je₁j je₁.0/j C 1

1

s 21

d11

(5.52)

Notingq1De1Cq1d,WO1 D QW1CW₁,kW₁k w1mand Assumption5.5, we obtain

jq1j je1j C jq1dj je1.0/j C 1 1

s 21

d11 C jq1dj 2L₁ k OW1k k QW1k C kW₁k

s 2₁

_min. ₁¹/Cw1m2L₁

Since the control1is a function ofr1andWO1, its boundedness is also assured.

q2- subsystem:

Similar to the analysis ofq1- subsystem, we have jr2j

vu

ut 2jd33j2

ˇˇˇd₂₂jd₃₃j d₂₃²ˇˇˇ; k QW2k s

22

min. ₂¹/ (5.53) Furthermore, we obtain

je2j je2.0/j C 1 2

vu

ut 2jd33j2

ˇˇˇd₂₂jd33j d₂₃²ˇˇˇ jq2j je2j C jq2dj je2.0/j C 1

₂ vu

ut 2jd₃₃j₂

ˇˇˇd₂₂jd33j d₂₃²ˇˇˇC jq2dj 2L₁ k OW2k k QW2k C kW₂k

s 2₂

min. ₂¹/Cw2m2L₁ (5.54) and thus the boundedness of control2.

q3- subsystem:

From the previous stability analysis about the q1-subsystem and the q2- subsystem, we know thatq1; q2; qP1; qP2are bounded. Accordingly, are bounded.

According to Lemma 5, we know that the internal dynamics are stable, i.e.,(q3

andqP3) are bounded. All the signals in the closed-loop system are bounded. This

completes the proof. ut

106 5 Altitude and Yaw Control of Helicopters with Uncertain Dynamics